With the combined analysis of synthetic aperture radar image and satellite altimeter data collected in the northeastern South China Sea (SCS), this study found one type of distorted phenomenon of internal solitary wave (ISW) with the long front caused by the oceanic mesoscale eddy. Motivated by these satellite observations, the authors carried out numerical experiments using the fully nonhydrostatic and nonlinear MITgcm to investigate the perturbation of ISWs by an isolated cyclonic or anticyclonic eddy. The results show that the ISW front is distorted by these oceanic eddies due to the retardation and acceleration effects at their two sides. The ISW energy along the front is focused onto (scattered from) the wave fragment where a concave (convex) pattern is formed, and the previously accumulated energy in the focusing region is gradually released after the ISW propagates away from the eddies. The ISW amplitude is modulated greatly by the eddies due to the energy redistribution along the front. Sensitivity results indicate that the magnitude of the modulated ISW amplitude in the focusing region can reach twice the incident ISW amplitude, while in the scattering region it can be reduced by more than a half. These results therefore suggest that models with eddies included, especially the energetic eddies, could further improve the amplitude predictions in the northeastern SCS. Additionally, the internal gravity wave formed behind the energy-focusing region by the anticyclonic eddies can steepen and break with the consequent formation of a secondary trailing ISW packet. Finally, this study shows that the model results of the distorted front and trailing packet are in qualitative agreement with that of the satellite observations in the northeastern SCS.
Numerous in situ and remote sensing observations provide evidence of large-amplitude internal solitary waves (ISWs) in the global ocean (Jackson 2004; Zheng et al. 2007). These waves, with their fronts extending for tens to hundreds of kilometers, carry a great amount of energy and can propagate several hundred kilometers away from their generation source sites (Liu et al. 2004; Vlasenko et al. 2005). The study regarding the evolution of these energetic ISWs is a topic of considerable significance. Although the shoaling ISW on the continental shelf/slope has been extensively studied in the past two decades (Lamb 2014), the evolution of ISW affected by the oceanic eddies is still not satisfactorily understood.
This study was motivated by a distorted phenomenon of ISW front (Fig. 1a) observed in the deep basin region in the northeastern South China Sea (SCS) by synthetic aperture radar (SAR) on 9 May 2001, during its propagation from the Luzon Strait (LS) to the Dongsha Island. Liu et al. (2004) suggested that this was due to the traditional refraction effect by bottom topography effect, which was similar to that around the Dongsha Island (Fett and Rabe 1977; Li et al. 2013; Guo and Chen 2014). Moreover, Zhang et al. (2011) suggested that the irregularity of source sites in the LS potentially resulted in a distorted ISW front. However, the cause of this distorted ISW front that propagated remotely from the LS might be much more complicated due to the variability in the location of remote generation sites and in transit times (Lamb 2014; Alford et al. 2015), which might be affected by various background mesoscale features (e.g., Buijsman et al. 2010; Park and Farmer 2013) and often strong barotropic tide and flow (e.g., Alford et al. 2010; Li and Farmer 2011). Except for the distorted ISW front, a small-scale trailing packet, that is, where the along-crest distance is much smaller than the leading transbasin wave, was also observed in Fig. 1, though with no obvious explanation [note, however, that the simulation by Zhang et al. (2011) suggested that the superposition of two wave fronts might provide one mechanism for forming this type of packet]. Also, satellite images show that the wave fronts generated near the LS tend to parallel to the line source sites, as predicted by Zhang et al. (2011). However, when the ISWs propagate away from the source sites and toward Dongsha Island, circular ISW fronts, which are different from the distorted phenomenon in Fig. 1, appear more easily in the deep basin because of the nonlinear effects (e.g., Liu et al. 2014; G.-Y. Chen et al. 2011; Zhao et al. 2004). Furthermore, because of the traditional refraction and diffraction effects, the distorted or cusped ISW fronts could be frequently observed and well simulated (e.g., Li et al. 2013; Zhang et al. 2011) after bypassing Dongsha Island. In summary, it can be seen that because of the variability in remote generation sites and in transit times, these unusual phenomena of ISWs in the SCS may be caused by many factors including tides, stratifications, generation sites, and the Kuroshio intrusions (Buijsman et al. 2010; Zhang et al. 2011; Park and Farmer 2013).
In addition to the above factors, on the basis of the satellite altimeter data, we found that a background anticyclonic eddy (AE) (Fig. 1b) actually appeared in the propagation process of these ISWs. This AE, which was shown from the geostrophic surface velocity anomaly (GSVA) and sea level anomaly (SLA), was detected on the same day with the ISWs observed by SAR. The magnitude of the GSVA induced by the AE reached up to ~39 cm s−1; however, the water depth here was more than 1000 m, and it was so deep that the traditional topography effect on the distortion of ISW front could be negligible (Grimshaw et al. 2004; Park and Farmer 2013). In this study, we expect that the background AE potentially plays a role in forming the distorted ISW front in Fig. 1. This type of phenomenon of ISW with a distorted front, which may be associated with background environments including not only the stratification but also the current, is called “distorted phenomenon” here to distinguish it from the traditional refraction phenomenon caused only by the medium (e.g., the changing stratification or/and bottom depth).
Eddies are also ubiquitous features in the Global Ocean (McWilliams 1985; Chelton et al. 2011), and the interaction between ISW and the eddy can be expected to form more easily when the two coexist. On the basis of the linear ray-tracing method, Park and Farmer (2013) demonstrated that in the northeastern SCS the Kuroshio intrusions could result in the advective refraction of ISWs and the deviation of predicted ISW arrival time (up to 2 h). However, the modulation of ISW characteristics, especially the ISW amplitude, by the oceanic eddies is still relatively unexplored, and more detailed and refined modeling studies will be helpful in quantifying the degree to which the eddies impact the location, speed, and amplitude of the ISWs.
Various weakly nonlinear theories based on the Korteweg–de Vries (KdV) equation were used with some success to model ISW evolution with long curved fronts. One of the most popular might be the Kadomtsev–Petviashvili equation (e.g., Pierini 1989; Cai and Xie 2010). This equation was modified by Grimshaw (1985) to include the weak two-dimensionality effect, where the across–wave front length scale was much smaller than the along–wave front scale. Another method to study the strong distortion of ISW fronts is the nonlinear geometrical optics approach. This method was modified by Small (2001) to study the refraction of ISW obliquely propagating across a slope or Gaussian seamount. However, the application of these weakly nonlinear theories (even when modified and extended) to the large-amplitude ISWs as that in Fig. 1 is under question because of the strong nonlinearity of the latter (Duda et al. 2004; Vlasenko and Stashchuk 2007, hereinafter VS07; Lien et al. 2012). In this study, in order to provide more precise quantitative information, the three-dimensional MITgcm (Marshall et al. 1997) that solves the fully nonhydrostatic and nonlinear equations is used to investigate how the ISW characteristics are perturbed by the oceanic eddies.
A better understanding of the role eddies play in modifying the ISW with the long front is essential to the prediction of ISW characteristics, especially the ISW amplitude, in the northeastern SCS (e.g., Simmons et al. 2011; Pickering et al. 2015). Motivated by this distorted phenomenon of the ISW front observed by satellite, this paper contributes to this understanding by reporting on numerical results obtained while investigating the effect of an isolated cyclonic eddy (CE) or AE on the perturbation of ISWs with long fronts. In section 2, we show the model setup and parameter choices. Next, results are presented and comparisons with satellite observations are discussed (section 3). Finally, section 4 presents the conclusions of this study.
2. Model description
The ISW with the long front affected by eddies similar to the scenario in Fig. 1 can be studied numerically under a sketch in Fig. 2, where an ISW with a straight-line front is assumed to propagate westward across an eddy (CE or AE). Next, the configurations of ISW and eddy and the parameter choices are shown.
a. Configurations of eddy and ISW in MITgcm
is used for the configurations of the eddy and ISW, where ψh and ϕ are the horizontal and vertical structures, respectively, of a single isolated eddy or ISW. The vertical structures of eddy ϕE and ISW ϕI can be given by solutions of the eigenvalue problems
respectively. Here, N(z) is the buoyancy frequency, f is the Coriolis parameter, H is the water depth, RE is the baroclinic Rossby deformation radius of eddy, and cI is the phase speed of linear internal waves. The horizontal structure of the eddy is given by
where M is the eddy intensity factor, and LE denotes the scale of eddy radius. Based on the KdV solution, the horizontal structure of ISW is given by
Here, parameters η0, V, and Δ are the amplitude, phase speed, and characteristic width, respectively, of the ISW in the KdV theory.
Next, the zonal velocity u and meridional velocity υ computed by u = −∂ψ/∂y and υ = ∂ψ/∂x, respectively, are used for initializing the eddy in the MITgcm; meanwhile, the zonal velocity u and vertical velocity w computed by u = ∂ψ/∂z and w = −∂ψ/∂x, respectively, are used for initializing the ISW (see below for details). Also, the sea surface change that is consistent with the surface flow and the isopycnal displacement that is in relation with the streamfunction are initialized into the MITgcm via the sea surface elevation field and the temperature and salinity fields, respectively.
b. Parameters in numerical experiments
In this study, in order to focus on the effect of mesoscale eddies on ISW evolution, a flat topography with a constant depth H = 1000 m is assumed. Moreover, a constant Coriolis parameter f = 5.23 × 10−5 s−1 corresponding to the latitude of 21°N is set. The oceanic stratification N(z) (Fig. 3a) calculated from the World Ocean Atlas 2009 (WOA09) data is used. The pycnocline depth h of the stratification is ~100 m, and the linear internal wave speed cI is ~2.1 m s−1. The vertical structures of the eddy ϕE and ISW ϕI are then calculated by solving the eigenvalue problems (2) and (3), respectively, as shown in Figs. 3b and 3c. Based on the KdV solution [(5)], an ISW is initialized in the MITgcm by setting the parameters η0 = 120 m, V = 2.35 m s−1, and Δ = 1160.4 m and then a new ISW with the wave amplitude η0 = ~102 m is formed after a period of adjustment; an adjustment process similar to this can also be seen in the works by Vlasenko et al. (2005). In this study, this new ISW is used as the standard incident wave unless otherwise stated.
For configuration of the eddy, the LE in (4) is set as 25 km. This scale is consistent with that of the first baroclinic Rossby deformation radius (20–40 km) at a depth of ~1000 m in the northeastern SCS (e.g., Cai et al. 2008) and is more than about 20 times larger than the ISW characteristic width. Besides, the statistical analysis shows that the eddies in the SCS have a lifetime of 7–10 weeks and a propagation speed of no more than 9 cm s−1 (G. Chen et al. 2011; Chelton et al. 2011). This implies that the mesoscale eddies in the SCS can be considered to be stationary in comparison with internal waves. The eddies are constructed by adjusting the eddy intensity factor M in (4). First of all, 25 numerical experiments (Table 1) are designed to investigate the effects of CE or AE on the perturbation of ISW fronts. When the parameter M > 0, the CE is constructed and its effects on the ISW are studied in six experiments (ECI1–ECI6); when M < 0, the AE is constructed and its effects on the ISW are studied in six experiments (EAI1–EAI6); and when M = 0, the experiment with only a single isolated ISW (EI) is constructed for comparison. Correspondingly, six experiments with only a single isolated CE (EC1–EC6) and six experiments with only a single isolated AE (EA1–EA6) are also run for convenience in the following analysis; in other words, the experiments EC1–EC6 and EA1–EA6 are run with only the eddy but without the ISW. The maximum surface velocity of the eddy in each experiment, denoted by Um, is also shown in Table 1. Similar to the parameter M, the positive (negative) of Um also represents that it is CE (AE) that is studied. For analytic purposes, the ratio of Um to the linear internal wave speed cI (i.e., Um/cI) is used to measure the relative intensity of eddies. In this study, the ratio is in the range |Um/cI| < 0.5, as shown in Table 1. Moreover, it is worth mentioning that although the water depth H is assumed as 1000 m in these experiments, the numerical results in this study can be scaled similarly to another ocean depth H1 by multiplying the LE and Um by H1/H (Dunphy and Lamb 2014).
To properly simulate the nonhydrostatic ISWs, Vitousek and Fringer (2011) showed that the horizontal grid resolution Δx must be less than the effective upper-layer depth he. Accordingly, the resolution Δx in the x direction between x = −160 km and x = 20 km is set as 100 m, and this gives a grid leptic ratio Δx/he that is less than 0.262 in all experimental cases (Here, the is no less than 381 m in all experiments.) This implies that the numerical dispersion is much smaller than the physical dispersion, and the nonhydrostatic behaviors related to ISWs can potentially be well produced by MITgcm. The resolution Δy in the y direction in the inner field between y = ±100 km, where the interaction between eddy and ISW happens, is also 100 m, while in the outer field it increases smoothly up to 500 m at y = ±170 km so that the inner field is not affected by the boundary signal in the whole process. The time step Δt is set as 12.5 s to satisfy the Courant–Friedrichs–Lewy condition (Vitousek and Fringer 2014), and the resulting maximum Courant number Cr = (V + |Um|)Δt/Δx, which is related to the internal gravity wave propagation, is smaller than 0.422 in all experimental cases. At the initial stage, the ISW with the wave front paralleling to the y axis is imposed at the east side of the domain (Fig. 2), and the eddy is centered at (xc, yc) = (−60 km, 0 km). In the z direction, there are 125 evenly spaced levels, giving a 8-m vertical resolution to capture the vertical structure of stratification N(z). Here, although the Orlanski radiation boundary conditions (Orlanski 1976) are used, they are not needed because the simulations are stopped before the waves reflect from the west boundary. Moreover, the free-slip conditions at the bottom and lateral boundaries are used. The values of horizontal and vertical viscosity are set as νh = νυ = 10−4 m2 s−1, and the values of horizontal and vertical diffusivity are set as Kh = Kυ = 10−5 m2 s−1.
3. Model results and discussions
In this section, experiments ECI3 and EAI3 with Um = ±0.68 m s−1 (Um/cI = ±0.324), which is close to the magnitude of the typical ocean eddies, for instance, observed in the SCS (e.g., Nan et al. 2011), are analyzed as the representative cases. Sensitivity analysis and comparisons with satellite observations are also presented and discussed.
a. Representative experiment ECI3
1) Modulation of ISW characteristics
Both the eddy and ISW can be distinguished from the isopycnal displacement, so the displacement of 1025 kg m−3 isopycnal (relative to a reference depth of about −250 m) in experiment ECI3 is shown (Fig. 4). The depression (elevation) ISW and AE (CE) result in the negative (positive) isopycnal displacement.
We start with the analysis in Fig. 4a (i.e., t = 2 h), which shows the result 2 h after the beginning. At this moment, because there is still quite some distance (~2.2LE) between the ISW and CE, the isopycnal displacement of the ISW changes little. At t = 5.5 h (Fig. 4b), the distance between the two decreases to ~1.0LE, and the eddy meridional velocity υ experienced by the ISW reaches maximum. However, still no obvious distortion happens to the ISW front, even though the isopycnal displacement at the center fragment of the ISW front becomes a bit weak.
The above results show that the modulation of the ISW front by the CE is weak at the time shown in Figs. 4a–b. Next, in Figs. 4c–f, the depression ISW is running into the dome-shaped pycnocline, and the effect of CE on ISW becomes strong. Based on the variations of background environments associated with the CE, the dynamical process of the modulation of ISW by the CE is divided into three stages for analysis purposes.
In stage I (x ≥ −60 km), the ISW is affected by the right part of the CE. In this stage, the ISW experiences an increasing zonal velocity u and a decreasing pycnocline depth. The result of the isopycnal displacement at t = 8.5 h after the evolution of this stage is shown in Fig. 4c, in which the ISW just arrives at the center of the CE (i.e., x = ~−60 km) and the ISW front is distorted obviously.
In stage II (−100 km ≤ x ≤ −60 km), the ISW has passed the eddy center and begins to be affected by the left part of the CE. In this stage, the experienced background environments by the ISW would be followed by a decreasing zonal velocity u and an increasing pycnocline depth. The result at t = 13.5 h (i.e., x = ~−100 km) after the evolution of this stage is shown in Fig. 4d, in which the distorted ISW front is further enhanced in comparison with that in Fig. 4c. In addition, the distribution of isopycnal displacement along the ISW front becomes quite uneven. This implies that the ISW energy along the wave front is greatly modified in this stage (see below for details).
In stage III (x ≤ −100 km), the ISW propagates away from the CE and the distance between them increases obviously, so the direct effect of CE on the perturbation of ISW ends and the ISW goes into a self-adjusting stage. In this stage both the above distortion and uneven distribution phenomena of isopycnal displacement of the leading ISW are gradually released. For instance, at t = 16.5 h (Fig. 4e) and t = 19 h (Fig. 4f), the ISW front near the place y = −25 km (i.e., −LE), which is short and strong at t = 13.5 h, becomes longer and weaker now due to the release of the previously accumulated energy along the wave front.
The results shown in Fig. 4 may be summarized as follows: First, a pattern of distorted ISW front is formed because of the retardation at the south side and acceleration at the north side of the CE; second, the isopycnal displacement of the leading ISW is unevenly redistributed because of the transport of energy along the distorted front. Next, we show the modulation of ISW characteristics, including wave amplitude η and phase speed c, in the x–z plane at three representatively vertical sections y = −25 (i.e., −LE), 0, and 25 km (i.e., LE). The section y = −25 km (y = 25 km) crosses through the south (north) side of CE; the section y = 0 km crosses through the center of CE. For the three sections, Fig. 5 shows the perturbations of the 1025 kg m−3 isopycnal (i.e., at z = −250 m), and Figs. 6a and 6b show the results of nondimensionalized ISW amplitude η divided by pycnocline depth h and nondimensionalized phase speed c divided by linear internal wave speed cI. Here, the phase speed c is computed by following the ISW peaks (i.e., c = dx/dt). Moreover, to make the isopycnal perturbations easier to see, Figs. 5g–i show the zoomed-in versions of Figs. 5d–f, respectively.
First, Fig. 5a (t = 2 h) shows that the isopycnal perturbations in the three sections are overlapped with each other, so the ISW amplitude η and phase speed c in three sections are almost the same (see the dots at x = ~−5 km in Figs. 6a,b). Next, at t = 5.5 h, the increase (decrease) of c in section y = 25 km (y = −25 km) becomes relatively obvious (see the dots at x = ~−35 km in Fig. 6b), while the difference in isopycnal perturbations (Fig. 5b) and ISW amplitude η (see the dots at x = ~−35 km in Fig. 6a) in three sections is still not very obvious. This suggests that the ISW amplitude η modulated by the CE experiences a delayed response in comparison with the phase speed c, which is similar to the case where ISWs propagate over a Gaussian seamount (Small 2001, see his Fig. 4). The reason for this type of delayed response is that the phase speed is directly affected by the background environments (e.g., the changing bottom depth, stratifications, and currents), while the ISW amplitude, whose variation is mainly a result of the focusing and scattering effects of wave rays (see below), is indirectly affected by background environments and thus its variation lags behind that of background environments and phase speed.
At t = 8.5 h, the ISW speed c (see the dots at x = ~−60 km in Fig. 6b) in section y = 25 km increases to the maximum, while in section y = −25 km it decreases to the minimum. The ISWs in three sections (Fig. 5c) propagate obviously one behind another. In detail, the ISW in section y = 25 km is in front, while in section y = −25 km it is left behind. This difference also leads to the distorted pattern of the ISW front in Fig. 4c. The ISW amplitude η (see the dots at x = ~−60 km in Fig. 6a) in section y = −25 km (y = 25 km) increases (decreases) to 1.1h (0.91h); and in section y = 0 km it only increases to 1.06h. However, different from c, we can see that in no section has the modulated η reached the maximum or the minimum until now.
At t = 13.5 h, because of the further effect of the CE in stage II, the isopycnal perturbation of the ISWs in three sections becomes more dramatic and complex, as shown in Figs. 5d or 5g. In section y = −25 km, the leading ISW is followed by a broad elevation internal gravity wave (named T), which is associated with the deformation of the CE (Fig. 4d). In section y = 25 km or y = 0 km, the leading ISW is followed by one depression ISW (named F), which is associated with a weak arc form front disintegrated from the leading ISW (Fig. 4d). Moreover, at this moment, the η (see the dots near x = ~−100 km in Fig. 6a) increases greatly up to 1.77h in section y = −25 km, while in sections y = 25 km and y = 0 km, it reduces to 0.73h and 0.68h, respectively. Also, we can see that in section y = −25 km, the η eventually reaches the maximum value at the end of stage II because of its delayed response in comparison with the c.
At t = 16.5 h and t = 19 h, another two ISWs named Ri (i = 1, 2) are also radiated at the two sides of the distorted ISW fragment near section y = −25 km, as shown in Figs. 4e and 4f. Because of the interaction with the radiated ISW R1, the ISW F in section y = 0 km is enhanced, especially at t = 19 h (Figs. 5f,i). Moreover, Fig. 6a shows that the η tends to decrease in section y = −25 km at these moments, and actually this is due to the self-adjustment effect in stage III through releasing the previously accumulated ISW energy along the front (see below).
The ISW phase speed c (Fig. 6b) in different sections is different because of the variations of background environments and/or wave nonlinearity, which then results in the distorted phenomenon of the ISW front in Fig. 4. Next, following Small (2001), the ISW phase speed c is further analyzed based on
where UE and NE are the background current and stratification associated with the mesoscale eddy; ϕ0 and k are the related modal structure and wavenumber; and f1(η) is the nonlinear part of c, and it is a function of ISW amplitude and background environments and increases with increasing absolute amplitude.
Figure 7 shows the modulations of c in two parts and reveals that the two parts modulate c in a quite different manner. The modulation by the linear part c0 is significant in stages I and II in comparison with stage III, while that by the nonlinear part f1 is significant in stage II and III in comparison with stage I. More specifically, in stage I and II, the two parts affect the ISW speed c in an opposite way. For instance, in stage I and II, because of the change of background environments in relation to the CE in section y = −25 km, the linear part c0 tends to decrease the c; on the contrary, because of the increase of ISW amplitude η, the nonlinear part f1 there tends to increase the c. Further, in stage III, the modulation of c by the linear part c0 basically ends, and only the nonlinear part f1 dominates the modulation. For instance, in stage III, Fig. 7 shows that the decrease of c (see Fig. 6b) in section y = −25 km is mainly due to the decrease of f1. Also, the decrease of f1 results from the decrease of η, as indicated in Fig. 6a.
2) Modulation of ISW energy
The above unevenly distributed ISW characteristics imply that the ISW energy should also be modified. Therefore, we analyze the variations of ISW energy. In each experiment, the total energy F(y, t), integrated along the x axis from xw = −160 km to xe = 0 km per unit length in y direction is written as
is the kinetic energy density and
is the available potential energy density. Here, g is the acceleration due to gravity, z*(x, y, z, t) is the height of the water parcel at (x, y, z) at time t in the reference state, and is the reference density (Lamb 2008; Lamb and Farmer 2011). The ISW energy E(y, t) along the front is computed approximately by deducting the energy of the eddy part from the total, and the energy of the eddy part is obtained by running the same eddy in isolation. For instance, using the result of eddy run in isolation in experiment EC3, we obtain the ISW energy E(y, t) along the front in experiment ECI3 by
Figure 8a shows the distribution of E(y, t) along the ISW front from t = 0 to 20 h. As stated above, the effect of CE on ISW starts from about t = 5.5 h (Fig. 4b), and therefore the ISW energy along the front is modified slightly from t = 5.5 h (Fig. 8a). Next, when the ISW begins to get away from the eddy center, clear hot and cold spots are gradually formed in Fig. 8a as a result of the strong focusing and scattering of ISW energy along the front. Furthermore, from t = ~13.5 h, when the ISW propagates away from the CE and enters into stage III, the previously accumulated ISW energy is gradually released from the focusing region to the scattering region. This type of focusing or scattering effect, which is due to the effect of the baroclinic eddy on ISW here, can also be found in other interaction cases (e.g., Dunphy and Lamb 2014; VS07). In Fig. 8a, the magnitude of the ISW energy in the focusing hot spot exceeds about twice the incident ISW energy, while in the scattering cold spot it reaches nearly zero, and it is found that this magnitude is comparable to the interaction case between internal tide and barotropic eddy under a constant buoyancy frequency (Dunphy and Lamb 2014).
Moreover, Fig. 8a reveals that, besides the bottom topography effect (e.g., the work by VS07), the eddy effect can also be significant to the redistribution of ISW energy along the front. And both the two effects (i.e., eddy and bottom topography) show that the ISW energy tends to be focused onto the wave fragment where a concave pattern is formed. For instance, VS07 suggested that under the bottom topography effect, the ISW energy was accumulated in the wave fragment near the region with prominent headland, where a concave wave fragment was formed because of the corresponding wave retardation effect there. Here, taking experiment ECI3 as an example, because of the retardation effect by the eddy, the ISW energy is focused on the south side where a concave fragment is formed similarly. Also, in this experiment, because of the acceleration effect by the eddy, a convex wave fragment is formed at the north side and the ISW energy is scattered from there. This scattering effect at the north side enhances the focusing effect at the south side.
3) Ray-tracing analysis
The above results indicate that the ISW energy is redistributed unevenly along the front because of the perturbation by CE. These focusing and scattering effects of ISW energy can be further illustrated by the ray-tracing analysis method, which has been adopted to examine the propagation behavior of different types of oceanic waves in different background environments (e.g., LeBlond and Mysak 1978; Kunze 1985; Park and Farmer 2013). For instance, by using the ray-tracing analysis, Kunze (1985) proves that the near-inertial wave energy tends to focus onto (scatter from) the location with a negative (positive) vorticity trough (ridge). Based on the similar ray-tracing analysis, a series of wave rays and fronts obtained under different background conditions associated with the CE in experiment ECI3 are displayed in Figs. 9a–c. For comparison, the initial wave front is also located near the east boundary of the computational domain, which is similar to that of experiment ECI3 in Fig. 4a. The wave rays are computed based on the linear wave speeds, which are obtained under different background conditions by solving the Taylor–Goldstein equation [(7)]. The combined effect of both mesoscale current and stratification on the pattern of wave rays and fronts is shown in Fig. 9a, while the separate effect of only the mesoscale current (stratification) is shown in Fig. 9b (Fig. 9c). Figure 9a shows that the wave rays are strongly focused onto (scattered from) the wave fragments where the concave (convex) wave fronts are formed. This phenomenon is consistent with the simulated results in Fig. 4, especially in stage I and II. However, the linear ray-tracing approach fails to reveal the simulated release effect in stage III (as seen in Figs. 4e,f) when the wave fronts propagate away from the CE. Anyway, the ray-tracing analysis results in Fig. 9a indicate that the CE can indeed lead to the distortion of wave fronts. Additionally, the results in Figs. 9b and 9c are helpful in distinguishing between the effect associated with changes in the wave speed due to the varying background current and effect of the varying background stratification. It is found that the variations of background current can result in the obvious acceleration and retardation of wave fronts at the two sides of the eddy (Fig. 9b), while the variations of background stratification only result in a weak retardation of wave fronts near the center area of the eddy (Fig. 9c). The ray-tracing analysis results in Figs. 9b and 9c imply that the distorted ISW fronts are mainly caused by the background mesoscale current variations, which account for ~83% of the overall distortion, while the effect of background stratification variations is relatively less important (only ~17%). Here, the overall distortion of the wave front is quantified by summing the distortions caused by mesoscale current and by stratification.
The effect of the background mesoscale current relative to that of the background stratification can be measured by an effective Froude number FrE defined as
where Δc0 is the change in the linear wave speed arising from the elevated or lowered pycnocline. The distortion of ISW fronts will be mainly caused by the background mesoscale currents if FrE > 1. In the representative experiment ECI3, FrE = ~6.74, and therefore the effect of the mesoscale current is strong relative to the stratification. Moreover, it is worth pointing out that the FrE ranges from ~5.6 to ~7 for all experimental cases, as shown in Table 1. Therefore, the weak effect of the background stratification relative to that of the background mesoscale current is generally true in this study.
b. Representative experiment EAI3
In experiment EAI3, because of the reversed rotation direction and the deepened pycnocline depth by the AE, the modulated results are different from that in experiment ECI3. One of the most obvious differences may be the reverse transport direction of ISW energy along the front. Figure 8b gives the redistribution pattern of the ISW energy E(y, t) for experiment EAI3 and shows that the transport direction of ISW energy affected by the AE is scattered from the south side and focused onto the north side. Figure 10 further displays the results of isopycnal displacement in experiment EAI3 at the latter four times (i.e., t = 8.5, 13.5, 16.5, and 19 h), when the modulated ISW front with distorted pattern is more obvious. In Fig. 10, corresponding to the energy-focusing and scattering effects in Fig. 8b, we can see that the modulated concave wave fragment is formed at the north side, while the modulated convex wave fragment is formed at the south side. Furthermore, the perturbations of the isopycnal in three representative sections as those in Fig. 5 are shown in Fig. 11. To make the isopycnal perturbations easier to see, Figs. 11e–g also show the zoomed-in versions of Figs. 11b–d, respectively. At t = 13.5 h (Figs. 11b,e), the modulated ISW amplitude η (163 m or 1.63h) with great increase appears in the north section y = 25 km and is smaller than the modulated value (177 m or 1.77h) in the south section y = −25 km in experiment ECI3. For easier comparison, the nondimensionalized ISW amplitude and phase speed in three sections are shown in Figs. 6c and 6d. Figure 6c shows that the modulated ISW with an increased η appears in the north section y = 25 km.
Although the modulated ISW characteristics and energy are reversed, the modulation process by the AE can also be divided into three stages, and the variation tendency of η (Fig. 6c) and c (Fig. 6d) in the three stages can also be analyzed similarly and comparatively with that in Figs. 6a, 6b, and 7. Moreover, similar to those in Figs. 9a–c, the focusing and scattering of ISW energy by AE can be studied by analyzing the ray-tracing results in Figs. 9d–f. However, because of the difference in stratification variations, the following results are worth mentioning.
First, the ISW phase speed c in each section in experiment EAI3 (Fig. 6d) is slightly larger than that which has a similar variation tendency in experiment ECI3 (Fig. 6b), especially at the place near the eddy center. For instance, at xc = −60 km, the c in section y = 25 km in experiment EAI3 is about 0.04 m s−1 (or ~0.02cI) larger than that in section y = −25 km in experiment ECI3. Actually, this difference is caused by the stratification variations associated with CE and AE and can be illustrated by the solution of (7). In experiment EAI3 (ECI3), the ISW runs into a bowl-shaped (dome shaped) pycnocline and the resulting deepened (shallowed) pycnocline makes the c increase (decrease) due to the increase (decrease) of its linear part c0. This opposite effect, which is caused by the difference in stratification variations, results in the slightly larger (smaller) value of c in experiment EAI3 (ECI3). Second, the ray-tracing results in Fig. 9f show that, in contrast to that in Fig. 9c, the stratification variations caused by the AE result in an acceleration effect of the wave fronts near the center area of the eddy. In Fig. 9f, the bowl-shaped pycnocline by AE makes the wave rays slightly scatter from the wave fronts near the center area of the eddy, while in Fig. 9c, because of the dome-shaped pycnocline by CE, the results are just the opposite.
Moreover, Figs. 10 and 11 show that three types of secondary waves are also formed in experiment EAI3. The disintegrated ISW F behind the scattering region and the radiated ISW Ri (i = 1, 2) at the two sides of the focusing region are similar to that in experiment ECI3. However, the broad elevation internal gravity wave T formed behind the focusing region has a different behavior in comparison with experiment ECI3, which can steepen and break with the consequent formation of a rank-ordered secondary trailing dispersive packet. Next, we focus on the formation of the trailing packet.
Secondary trailing ISW packet
In experiment ECI3, no breaking happens to the elevation internal gravity wave T (Fig. 5f, 12a). However, in experiment EAI3, the broad elevation internal gravity wave T steepens at first (Figs. 11b or 11e), and eventually it disintegrates into a trailing ISW packet at its rear shoulder (Figs. 11g or 12b). Recently, Grimshaw et al. (2014, hereinafter G14) simulated another type of trailing ISW packet generated by the combined effect of rotation and topography in the northern SCS. On the basis of the Ostrovsky equation and simulations, they found that the rotation effect tended to enhance the trailing inertial-gravity wave steepening and breaking as long as the evolution distance was large enough. Here, in order to illustrate the difference in behavior of the trailing internal gravity wave T between experiments EAI3 and ECI3, the Ostrovsky number Os, defined as (Grimshaw et al. 2012; G14)
and the Ursell number Ur, defined as (e.g., Farmer et al. 2009)
are calculated for the two experiments (Table 2), where ηs, Ls, and κ represent the amplitude, horizontal scale, and the maximum curvature of the trailing wave T, respectively, as marked in Fig. 11e; α, β, and γ are the coefficients of the quadratic nonlinear term, linear dispersive term, and rotational dispersive term of the Ostrovsky equation (see G14 and Table 2 for details).
It was suggested that the breaking and formation of ISWs were much easier to achieve (Farmer et al. 2009; Grimshaw et al. 2012) if the nonlinearity dominated over the linear dispersion and rotation dispersion, that is, Ur > 1 and Os > 1. Here, Table 2 shows that the ηs of the trailing wave T in experiment EAI3 is much larger than that in experiment ECI3 because of the downward deepening and upward shallowing of pycnoclines by AE and CE, respectively. Then, it is seen that both the Ursell and Ostrovsky numbers (Table 2) in experiment EAI3 are much larger than that in experiment ECI3 according to Eqs. (13) and (14). Therefore, the trailing wave T in experiment EAI3 with a much stronger nonlinearity is much easier to steepen and break with the consequent formation of ISW packet, as reflected in Fig. 12b. Moreover, because of their reduced length scale (2–4 km) and amplitude (~30 m), the Ursell number of these high-frequency ISWs decreases to ~5, which shows that a balance between nonlinearity and nonhydrostatic dispersion tends to be reached eventually.
The above results suggest that the formation of the trailing ISW packet here needs a finite trailing wave amplitude ηs, which depends on the intensity of the AE. To further illustrate this point, the result (at t = 19 h) in experiment EAI5 with a weaker eddy intensity than experiment EAI3 (Table 1) is shown in Fig. 12c, in which both the number of ISWs and the amplitude of leading ISW in the trailing dispersive packet decrease obviously because of the decrease of ηs. These variations thus show that the change of ηs has the determinative effect on the nonlinear strength of trailing internal gravity wave T and the eventual evolution of the trailing dispersive packet. It is noteworthy that in G14, although the ηs of the trailing inertial-gravity wave has no change with η0, the trailing wave nonlinearity and evolution are still affected because of the change of Ls, as implied by
Here, Q is the linear magnification factor, as seen in G14. Equation (15) shows that the horizontal scale Ls actually depends on η0 and thus, according to (14), its change will have the determinative effect on the nonlinearity of the trailing inertial-gravity wave and then the evolution of the trailing dispersive packet. This significant difference in effect on trailing wave nonlinearity and evolution shows that the trailing wave here is essentially different from that in G14, although both of the two types of trailing waves are able to eventually evolve into nonlinear ISW packets.
c. Sensitivity analysis and comparison with satellite observations
The results presented above show how the CE or AE affects the ISW characteristics and energy redistribution along the front. Obtained for Um = ±0.68 m s−1 (Um/cI = ±0.324), the variation tendency can actually be valid in a wide range of eddy intensities. Here, the disturbance of the ISW energy , expressed as
is used for the sensitivity study to different eddy intensities, and the positive (negative) disturbance represents energy focusing (scattering). Figure 13 displays the sensitivity of to different eddy intensities at t = 13.5 h, when the typical focusing and scattering effects are very obvious (sections 3a and 3b). Figure 13 shows that the ISW in the experiment with stronger eddy intensity has a larger energy disturbance, and the energy transport direction along the front modulated by CE (AE) is still scattered from the north (south) side and focused onto the south (north) side in a wide range of eddy intensities.
Next, we show the sensitivity of the modulated ISWs with maximum amplitude to the intensity of eddies. Figure 14a shows the normalized maximum amplitude ηm/η0 versus Um/cI, and Fig. 14b depicts how the normalized location (Xm/LE, Ym/LE) of the maximum amplitude moves relative to the center of the eddy. Here, ηm represents the modulated maximum amplitude appearing at (xm, ym); Xm/LE [i.e., (xm − xc)/LE] and Ym/LE [i.e., (ym − yc)/LE] represent the normalized displacements of ηm in the x and y axes, respectively. For instance, in experiment ECI3, the ηm reaches up to 183 m, which is about 1.8 times the incident ISW amplitude η0 (i.e., ηm/η0 = ~1.8; Fig. 14a). Moreover, the ηm in experiment ECI3 appears at the southwest of the CE with (xm, ym) = (−96.1 km, −20.2 km), and thus its normalized location is (Xm/LE, Ym/LE) = (−1.44, −0.81) (Fig. 14b). Similarly, in experiment EAI3, it shows that the modulated ISW with maximum amplitude (ηm = 172 m, i.e., ηm/η0 = ~1.7; Fig. 14a) appears at the northwest of the AE with (xm, ym) = (−105.1 km, 23.9 km) [i.e., (Xm/LE, Ym/LE) = (−1.8, 0.96); Fig. 14b].
In summary, the results in Fig. 14a show that the ηm increases with increasing absolute eddy intensity, and it can reach twice the incident ISW amplitude in the focusing region. On the other hand, in the scattering region the modulated ISW amplitude can be reduced by more than a half (as seen in Figs. 6a,c). Moreover, Fig. 14b shows that in the x axis the |Xm| modulated by both AE and CE is larger than the eddy radius LE (i.e., |Xm/LE| > 1) and decreases with increasing absolute eddy intensity; however, in the y axis, the |Ym| modulated by both AE and CE is smaller than LE (i.e., |Ym/LE| < 1) and increases with increasing absolute eddy intensity. Therefore, we conclude that the location of ISW with maximum amplitude modulated by eddies with relative intensity |Um/cI| < 0.5 generally appears at the end of stage II or the beginning of stage III (Fig. 14b), when/where the direct effect of the eddy on ISW is nearly coming to an end.
In addition to the above sensitivity analysis to eddy intensity, it is also useful to explore the sensitivity of the evolution results to the incident ISW amplitude η0. Here, based on representative experiment ECI3, another two experiments ECA1 and ECA2 are performed by modifying η0 as 39 and 70 m, respectively. Note that the three experiments with different η0 are run with the same Um (=0.68 m s−1). Figures 15a–c show the displacements of the 1025 kg m−3 isopycnal in the three experiments at t = 19 h and reveal that a similar pattern of distorted front appears in a wide range of incident ISW amplitudes because of the similar focusing and scattering effects of the ISW energy along the different fronts. However, Figs. 15a–c emphasize that the modulated ISW font in the experiment with a larger value of η0 has a broader focusing region. This can also be seen further in Fig. 15d by investigating the sensitivity of to η0. Figure 15d shows that while increasing by a similar magnitude at about 1.2–1.4 times in all three experiments with different η0, the energy of ISW in the experiment with larger η0 is focused onto a broader focusing region. For instance, the width of the focusing region in experiment ECI3 with the largest η0 is about 1.7 times broader than that in experiment ECA1 with the smallest η0.
Finally, on the basis of our numerical results, we discuss the satellite-observed ISW characteristics in Fig. 1. This SAR-observed ISW with distorted front, which is different from the most commonly observed circular ISW front (e.g., Zhao et al. 2004; Liu et al. 2014) in the deep basin region in the northeastern SCS, is shown together with a background AE observed by satellite altimeter on the same day. According to the relative position between the AE and the long ISW fronts, it can be deduced that the effect of the eddy on the perturbation of ISWs is currently at stage III. The position of the concave wave fragment in Fig. 1 just appears at the northwest of the AE, which is qualitatively consistent with the simulated characteristic of the long ISW front in Figs. 10b or 10c. In Fig. 1, the magnitude of the maximum surface velocity Um of the AE in a water depth of about 3 km is only ~0.39 m s−1, which is much weaker than that in Fig. 10. Therefore, the degree of distortion of the long ISW front in Fig. 1 is much weaker than that in experiment EAI3. Moreover, in Fig. 1, a rank-ordered ISW packet with a relatively short front (thin red lines) is found to be propagating at the northwest side of the AE. The propagation direction (denoted by the thick black arrow in Fig. 1) of this SAR-observed trailing ISW packet is toward the northwest, which is also qualitatively consistent with that of the simulated trailing ISW packet T in Figs. 10 and 11.
In this paper, the fully nonhydrostatic and nonlinear MITgcm has been used to perform high-resolution, three-dimensional numerical simulations of ISW interactions with mesoscale eddies in the northeastern SCS. The dynamical process, characteristics, energy redistribution, and trailing wave packet of the modulated ISW are studied.
The numerical results show that the dynamical process of the modulated ISW can be divided into three stages in terms of the relative position between ISW and eddy. In stage I, the ISW is affected by the right part of the oceanic eddies, and its front is distorted obviously after the perturbation of this stage. In stage II, the ISW is affected by the left part of the oceanic eddies. In this stage, the distorted ISW front is further enhanced and the distribution of the ISW characteristics and energy along the wave front becomes quite uneven. In stage III, the ISW is assumed to have propagated away from the oceanic eddies, and it goes into a self-adjusting stage. In this stage, the direct effect of eddies on ISW evolution ends, and the uneven distribution phenomena of the ISW characteristics and energy are gradually released.
This distorted ISW front is mainly due to the acceleration and retardation effects at the two sides of the eddy. The modulation of the ISW phase speed is dominated by the linear part when the ISW propagates through the eddy, while the nonlinear part becomes significant when it passes the eddy. The modulated ISW energy tends to be accumulated in stage I and II, while in stage III, when the ISW propagates away from the eddy, the previously accumulated energy will be gradually released. The ISW energy along the front affected by the eddy is scattered from (focused onto) the wave fragment where a convex (concave) pattern is formed. From the standpoint of predicting the ISW characteristics in the SCS, the distortion of ISW fronts by mesoscale eddies has a weak impact on the accuracy of the predicted ISW arrival time (or on the total propagation time) because the scale [O(1–10) km] of distortion in the across–wave front direction is much shorter than that of the total propagation distance [O(100) km] in the deep SCS basin (Liu et al. 2004; Zhang et al. 2011). However, in this study, the results show that the ISW amplitude is greatly modified by the mesoscale eddies due to the energy redistribution along the distorted ISW front. Therefore, this study suggests that the models with eddies included, especially the energetic eddies, could further improve the accuracy of the ISW amplitude predictions, although the arrival time of the ISWs in the northeastern SCS has been predicted to a reasonable degree of accuracy with models that omit the mesoscale currents (e.g., Zhang et al. 2011; Simmons et al. 2011; Park and Farmer 2013).
On the basis of the ray-tracing analysis, we show that the distorted ISW fronts are mainly caused by background mesoscale current variations, while the effect of background stratification variations is relatively less important. In the case with a CE (AE), the ISW runs into a dome-shaped (bowl shaped) pycnocline, and the resulting shallow (deepened) pycnocline makes the phase speed c decrease (increase). This opposite effect on the phase speed is caused by the difference in stratification variation, which makes the wave rays focus onto (scatter from) the fronts near the center area of CE (AE) and results in the corresponding retardation (acceleration) of wave fronts there.
Moreover, the sensitivity analysis to a wide range of eddy intensities and incident ISW amplitudes is performed. The sensitivity results show that the magnitude of the modulated ISW amplitude in the focusing region can reach twice the incident ISW amplitude, while in the scattering region it can be reduced by more than a half. The location of ISW with maximum amplitude modulated by eddies with relative intensity |Um/cI| < 0.5 generally appears at the end of stage II or the beginning of stage III, when/where the direct effect of the eddy on ISW is nearly coming to an end.
In addition to modulated results associated with the long distorted leading front, another interesting feature is the formation of the trailing-like ISW packet with a relatively short wave front. It is found that the internal gravity wave formed behind the energy-focusing region by the AE can steepen and break with the consequent formation of the secondary trailing ISW packet. The number of ISWs and the amplitude of leading ISW in the trailing dispersive packet are quite sensitive to the intensity of the AE, and they increase with the trailing internal gravity wave amplitude ηs. The comparisons between the model results and satellite observations show that both the simulated pattern of the distorted ISW front and radiation of the secondary trailing packet can be in qualitative agreement with the satellite observations in the northeastern SCS. Thus, this study shows that the eddy effect potentially provides a mechanism for the formation of the distorted ISW front and trailing wave packet.
The authors gratefully acknowledge Oliver Fringer and the other anonymous referee for providing numerous comments that significantly improved the quality and clarity of the manuscript. This work was jointly supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (XDA11020201); the National Basic Research Program (2011CB013701 and 2013CB956101); NSFC Grants 41406025, 41206009, 41406019, 41430964, 41025019, and 41521005; the CAS/SAFEA International Partnership Program for Creative Research Teams (20140491532); and the Innovation Group Program LTOZZ1502 of State Key Laboratory of Tropical Oceanography (South China Sea Institute of Oceanology, Chinese Academy of Sciences). The numerical simulations were performed on the HPCC at the South China Sea Institute of Oceanology, Chinese Academy of Sciences. The altimeter products were produced by Ssalto/Duacs and distributed by AVISO, with support from CNES (http://www.aviso.oceanobs.com/duacs/).