Strongly Nonlinear Effects on Determining Internal Solitary Wave Parameters From Surface Signatures With Potential for Remote Sensing Applications

The inversion of remote sensing signatures of internal solitary waves (ISWs) can retrieve dynamic characteristics in the ocean interior. However, the presence of ubiquitous large‐amplitude ISWs poses challenges to the commonly used weakly nonlinear methods for parameter retrieval. Through laboratory experiments, we establish a relationship between surface features and internal characteristics of ISWs by the remote sensing imaging mechanism. The results demonstrate that strong nonlinearity significantly influences the retrieval of ISWs, primarily manifested in the calculation of wave‐induced velocities and the applicability of ISW solutions. A fully nonlinear model Dubreil–Jacotin–Long equation is used in the retrieval and has been tested under different conditions. Mooring observations indicate that the determination of ISW parameters from satellite images is affected by the complexity of in situ stratification, but additional remote sensing information such as surface velocities enables us to perform retrievals even if the real‐time measurement of pycnocline depth is not available.

Supporting Information may be found in the online version of this article.

RESEARCH LETTER
been discussed under various conditions such as wind, surface waves, and surfactant films (Brandt et al., 1999;Craig et al., 2012;da Silva et al., 1998;Romeiser et al., 1997;Wen et al., 2022;Yue et al., 2023).(b) The modulation effect of ISWs on surface currents.In a framework where the scattering intensity is proportional to the ISW induced surface velocity divergence (Alpers, 1985;Jackson et al., 2013), ISWs and surface currents are typically modeled using different ISW theories in the retrieval.
However, models based on the WNL assumption are inadequate to describe strongly nonlinear (SNL) ISWs (Grue et al., 1999(Grue et al., , 2000)), which account for a large proportion of ISWs in in situ oceanic observations (Chang et al., 2021;Huang et al., 2022;Stanton & Ostrovsky, 1998;Yang et al., 2021).Although WNL theories have been used outside their formal range of validity in some cases (Helfrich & Melville, 2006), for a detailed description of the dynamics of large-amplitude ISWs observed in the real ocean, theoretical models with strong nonlinearity are needed.The Miyata-Choi-Camassa (MCC) equation, a fully nonlinear equation of long waves was proposed and agreed well with laboratory experiments (Camassa et al., 2006;Kodaira et al., 2016).Meanwhile, without any assumptions about wavelength and amplitude, the fully nonlinear Dubriel-Jacotin-Long (DJL) equation has been applied effectively in the laboratory and the ocean (Camassa et al., 2018;Lien et al., 2014;Luzzatto-Fegiz & Helfrich, 2014).To date, strongly nonlinear effects on the retrievals of wave parameters have not been fully considered in the following two aspects: (a) the calculation of wave-induced velocities and (b) the applicability of ISW solutions.To a certain extent, these deficiencies restrict the accurate determination of ISW parameters, the adequate modeling of ISW dynamics should be determined in the retrieval (Romeiser & Graber, 2015).
To address the difficulty of matching mooring observations and satellite images of the same ISW, this paper establishes the relationship between surface features and internal characteristics of ISWs in laboratory experiments.Then the strongly nonlinear effects on retrieving ISW parameters are evaluated in terms of wave-induced velocities and ISW solutions.Finally, we assess the retrieval of wave parameters in oceanic environments.

Laboratory Experiments and Data Processing
Experiments are conducted in the stratified tank of the Key Laboratory of Physical Oceanography, Ocean University of China (schematic diagram is shown in Figure 1d).The tank is 6 m long, 0.25 m wide, and 0.5 m tall, where two layers of fluid with thicknesses of h 1 (upper layer) and h 2 (lower layer) and densities of 1,020 and 1,040 kg/m 3 are injected respectively.The depth ratio of the lower and upper layers h 2 /h 1 vary from 3 to 10, see Table S1 in Supporting Information S1 for detailed conditions.A conductivity probe is driven vertically by a stepper motor to measure the stratification.The fluid is seeded with 40-μm-diameter tracer particles made of polystyrene and illuminated by two lasers (532 nm).The wave-induced velocity is measured by particle image velocity (PIV) (Thielicke & Stamhuis, 2014).A textured mask emitting light at a wavelength of 400-480 nm is placed behind the tank.The waveform η(x) is determined by identifying the depth where the maximal vertical density gradient is located using synthetic schlieren (SS) technique (Dalziel et al., 2007).Two synchronous charge coupled device (CCD) cameras with bandpass light filters are set in front of the tank and recorded at a frequency of 40 Hz, each CCD has a horizontal field of view of 1.25 m and they simultaneously record PIV and SS images respectively, as shown in Figure S1 in Supporting Information S1.The waves are generated by the lock-release method (Kao et al., 1985) using a movable gate of the same width as the tank.The non-dimensional amplitudes η 0 /h 1 ranging from 0.18 to 2.50, where η 0 is defined as and the wavelength of an ISW is defined as For the problem we raised, it is enough to consider the influence of surface currents on remote sensing signatures in the framework of weak wind wave-current interaction and Bragg scattering theory, which can be expressed as (Alpers, 1985;Jackson et al., 2013) where ∆σ = σ − σ 0 denotes the deviation of the normalized radar cross-section intensity from its mean value, u s denotes the velocities of surface currents, and the x direction is defined as the direction of wave propagation.This means that the surface manifestations of ISWs (Figures 1a and 1b) can be calculated from the divergence of surface velocities (Figure 1c) in the laboratory.In satellite images, the distance between the positive peak and the adjacent negative peak D p−p of ISWs (Figure 1b) is less affected by winds in most cases compared to the signal intensity, hence the peak-to-peak distance D p−p is widely chosen to characterize the remote sensing signature of ISWs (Xue et al., 2013;Zheng et al., 2001).In our experiments, it can be expressed as (4) In the laboratory,  − is one to two orders of magnitude larger than the scale at which surface tension becomes significant (Holthuijsen, 2010).Therefore, the impact of surface tension on the  − can be neglected.

Weakly Nonlinear Theories
For ISW solutions under the WNL assumption, we estimate KdV, JKKD, BO, and eKdV equations (see Text S1 in Supporting Information S1).
The wave-induced velocity used for quantifying surface divergence can be determined by the stream function (Stastna & Peltier, 2005): where η(x,z) is the wave-induced vertical excursion of isopycnals.Under the WNL assumption, for a given waveform η(x), the solution of wave-induced horizontal velocity has the following separable form, in which the vertical modes are independent of the horizontal location (Benney, 1966): where c 0 is the linear phase speed and ϕ b (z) is obtained by solving the Sturm-Liouville boundary value problem, written as: where N b (z) is the given background buoyancy frequency, calculated by: where g is the gravitational acceleration, ρ(z) is the density profile, and ρ 0 is the reference density.

Strongly Nonlinear Theories
For SNL theories, we consider ISW solutions of MCC and DJL equations (see Text S1 in Supporting Information S1).
If strong nonlinearity is taken into consideration, the displacement of streamlines induced by ISWs should be accounted for.Therefore, the vertical structure function needs to be calculated in the presence of wave motion and depends on x as well (Apel, 2003).The solution of the induced horizontal velocity is calculated in the following inseparable form: where ϕ wave (x,z) is the vertical structure function when the wave exists.A simple iterative method is proposed to calculate ϕ wave by iterating the changed stratification and η(x,z) with the initial value of η(x) (see Text S2 and Figure S2 in Supporting Information S1).

Strongly Nonlinear Effects on the Solution of Wave-Induced Velocity
Figures 2a-2c show comparisons between theoretical and experimental wave-induced velocities of the same ISW.We consider an example with h 2 /h 1 = 5 and η 0 /h 1 = 1.44,where the amplitude is of the same magnitude as h 1 and shows significant nonlinearity.The waveform (black solid lines in Figures 2a and 2b), phase speed, and stratification for calculating the theoretical wave-induced velocities are given by the experiment.Compared to the experimental and SNL theory (Equation 10, Figure 2b) results, the WNL theory (Equation 7, Figure 2a) exhibits deviations in both magnitude and structure, similar to the result of Grue et al. (1999), more details are shown in Figure S3 in Supporting Information S1.In the upper layer, the velocity solution of the WNL theory is more than twice the SNL theory, this difference between measurements and WNL theory was also observed in the ocean, see Figure 13 of Rong et al. (2023).Consequently, from the perspective of surface velocity calculations, the SNL theory yields a larger D p−p than the WNL theory for the same ISW (Figure 2c).This difference reaches a maximum of 33% with increasing of η 0 in the scope of our experiments.In Figure 2d, theoretical D p−p of each experimental waveform is calculated and compared to measurements.Prior studies suggested that the relationship between D p−p and L w can be obtained directly using WNL theories, such as D p−p = 1.32 L w in the KdV equation, and the ratio is independent of amplitude (Small et al., 1999;Zheng et al., 2001).However, the SNL theory shows that the ratio will reach 2.7 with increasing amplitude, which is consistent with experimental results.It implies that the wavelength determined from remote sensing is typically overestimated for ISWs with strong nonlinearity.Furthermore, only when the waveform is weakly nonlinear, D p−p /L w can be a constant.The differences in ISW solutions will be shown in Section 3.2.

Remote Sensing Characteristics of ISWs
Combining the SNL velocity calculation method used in Section 3.1 and ISW solutions, the relationship between remote sensing characteristics and wave parameters will be established in this section.Here we take cases of Figures 3a-3c show the variation in wave parameters with the peak-to-peak distance D p−p .The results from the KdV, JKKD, and BO equations show a similar pattern, in which parameters change monotonically with D p−p .For the eKdV, MCC, and DJL equations, the relationship between the wave parameters and D p−p is no longer monotonic, and significant differences are exhibited when amplitudes become large.Its typical feature is the existence of a turning point, which means that one D p−p will correspond to two parameters, that is, the existence of double solutions.The selection method for double solutions can be determined by the properties of wave packets that the leading wave reaches the maximum amplitude (Xue et al., 2013).The turning points can also be directly distinguished if the phase speed is measured through remote sensing, such as using satellite image pairs (Liu et al., 2014) and X-Band radar (Celona et al., 2021).The positions of turning points in different stratification are shown in Figure S4 in Supporting Information S1, indicated by c/c 0 = 1.08-1.16.
To quantitatively compare the differences between ISW theories and experimental results in retrievals, we define the relative deviation (RD) as follows where λ theory and λ exp are any theoretical and experimental parameters under a specific D p−p .The results of the comparisons under different amplitudes and h 2 /h 1 indicate that the assumption of waves limits the application of ISW solutions in retrievals (Figure S8 in Supporting Information S1).Solutions under the WNL assumption prove inadequate in describing large-amplitude ISWs, and the long wave assumption leads to significant deviation in the MCC equation when the depth increases.The DJL equation demonstrates broad applicability across various conditions, with RD of parameters typically below 10% in laboratory experiments.
Image intensity, another information of ISWs obtained from satellite images, is partly modulated by the divergence of u s (Equation 3).Figures 3d and 3e show that both the maximum values of u s and its divergence are directly proportional to the amplitude of ISWs (Vlasenko et al., 2000), which means that under the same oceanic conditions, the brighter or darker the stripes in the image indicate the larger amplitudes.However, the image intensity is sensitively influenced by environmental factors such as wind and surface waves (Magalhães, Alpers, et al., 2021), making it difficult to be widely quantitatively applied.Qualitative analysis (Magalhães & da Silva, 2012;Magalhães, Pires, et al., 2021) and more detailed sea surface dynamical modeling about the intensity will facilitate the acquisition of additional information from remote sensing.

Discussion
The above work explores the retrieval of wave parameters in a quasi-two-layer procedure under laboratory conditions.However, complex stratification in the ocean brings difficulties and variety to retrieval.Therefore, the applicability of the established method is further tested with mooring observations.The thickness of the pycnocline which spans several hundred meters in deep seas affects the solution of ISWs (Allshouse & Swinney, 2020;Fructus & Grue, 2004).Figure 4a shows DJL solutions at various depths corresponding to the locations of mooring stations.The 200 m depth station is S5 during the Asian Seas International Acoustics Experiment (ASIAEX) (Duda et al., 2004), the 600 m depth station is LR1 of Chang et al. (2021), the 953, 2,762, and 3,745 m depth stations are M1, M6, and M10 of Huang et al. (2022).The stratification used for the DJL equation is given by the yearly mean WOA18 data set at each location except for the station of 200 m, where the stratification is given by the average value of temperature chains during ASIAEX.The curves below the measured maximum amplitude (dashed lines) show different patterns.As D p−p increases, the amplitudes in the solutions show a roughly monotonic decrease at stations with depths exceeding 953 m, indicating that the amplitudes above the turning point no longer need to be considered, simplifying the retrieval in deep seas.Meanwhile, the amplitudes below and above the turning point still require consideration in shallow seas.
For deep seas, the relatively stable stratification provides us with more possibilities for accurate retrieval.However, the pycnocline in shallow seas appears to be variable.In Figure 4b, 36 ISWs in S5 were counted during ASIAEX.The amplitude η 0 is defined as the displacement of 24°C isotherm (Ramp et al., 2004).Using the time series of upward-looking ADCP with an interval of 2 min, we estimate D p−p by where Vel s is the wave-induced velocity near the sea surface (37 m).Background currents are calculated by averaging velocities 30 min before the arrival of ISWs and then removed.The divergence of Vel s of an ISW is shown in Figure S9 in Supporting Information S1, and it matches well with a corresponding SAR image (Liu et al., 2004), providing support for further analysis using D p−p .The observations show that even in 15 days, the upper layer depth will vary between 30 and 60 m, which may be caused by internal tides and surface forcing (Font et al., 2022).The observed ISWs are scattered around the DJL solutions calculated with the average stratification (the blue line), approximately distributed within a range calculated by the DJL equation with h 1 = 30 and 60 m.The amplitudes are affected by this variation in stratification in a few days, as observed by Small et al. (1999) and Lien et al. (2014).Although applying the DJL equation with average stratification in an area allows for some statistical analysis of ISWs using satellite images, the difficulty in obtaining real-time stratification may introduce uncertainties in retrieving the parameters of a specific ISW.
Previous studies have explored the feasibility of using SAR images to estimate the pycnocline depth (Dessert et al., 2022;Le Caillec, 2006;Zhao et al., 2004).However, the requirements for the specific process (polarity conversion) or quantitative modeling of image intensity limit the widespread application of these methods.Are there other available remote sensing variables to avoid the impact of the unknown stratification in the retrieval?Compared to image intensity determined by velocity divergence and other oceanic factors, velocity appears to be a more suitable choice, which monotonically changes with η 0 (Figure 3d).Some new technologies for measuring surface currents induced by ISWs have been developed, such as interferometric SAR (Romeiser & Graber, 2015) and airborne PIV (Vrećica et al., 2022).If the maximum velocities of ISWs and D p−p are both taken into account, a new accurate retrieval method can be established.DJL solutions under different h 1 are shown in Figure 4c, with a density difference between the upper and lower layers of 3.3 kg/m 3 (the variation is minor during the observation) and a total depth of 200 m.It indicates that one D p−p and one maximum Vel s can uniquely determine the amplitude of an ISW, even when the pycnocline depth is unknown.Using this method to retrieve the 36 ISWs in Figure 4b and the results are presented in Figure 4d (note that the D p−p in Figure 4c is transformed into D p−p /c during this retrieval due to the absence of the phase speed in mooring observations).The root mean square error (RMSE) between the retrieval results and the observations decreases significantly from 16.8 m (Figure 4b) to 9.6 m in Figure 4d.The retrieval results of large amplitude ISWs show an underestimation, which is caused by the deformation (marked by red squares) due to the steepening of the ISWs (Sutherland et al., 2013).More investigations are required to estimate the shoaling processes of ISWs from remote sensing.

Conclusions
In this study, we established the dimensionless relationship between the surface and internal characteristics of ISWs in laboratory experiments.Different from weakly nonlinear theories commonly used for retrieval, strong The relationship between D p−p and the amplitude of ISWs is monotonically decreasing in deep seas but not in shallow seas due to the influence of the pycnocline thickness.The determination of ISW parameters from satellite images in oceans is affected by the variation of in situ stratification.The integration of a diverse set of remote sensing techniques helps us acquire additional information about ISWs.The distribution of stripes in satellite images along with measurements of sea surface velocities enable accurate retrieval even when the pycnocline depth is unknown.
This work provides a validated model for determining the parameters of steady ISWs from surface signatures with potential for remote sensing applications.Moreover, ISWs can be modulated by background currents, especially shear currents (la Forgia & Sciortino, 2021;Stastna & Lamb, 2002).The difficulty in measuring real-time currents in the ocean poses new challenges for the retrieval of ISWs.Further research will be conducted to investigate the interaction between ISWs and other oceanic processes.The study was supported by the National Natural Science Foundation of China through Grant 41876015 and the National Key Research and Development Program of China through Grant 2021YFC3101603.The authors would like to thank the editor and two anonymous reviewers for their efforts in improving this paper.We also thank Dr. Michael Dunphy for the DJL solver (https://github.com/mdunphy/DJLES).
features and internal parameters of internal solitary waves is established in laboratory experiments • Strong nonlinearity significantly impacts the determination of wave parameters from the surface.A fully nonlinear model is well applied • Accurate parameter determination is constrained by the complex oceanic stratification, but more remote sensing information can overcome it Supporting Information:

Figure 1 .
Figure 1.Schematic of experiments.(a) ISWs captured by ENVISAT on 18 June 2008, UTC.(b) Relative image intensity along the direction of wave propagation (the red line in a).(c) The surface horizontal velocity divergence of the ISW in (d).The gray and black lines present the original and low-pass filtered results, respectively.Red dashed lines indicate the horizontal position corresponding to the maximum and minimum values.(d) Schematic diagram of the experiment.

Figure 2 .
Figure 2. (a)-(c) Comparisons between theoretical and experimental velocities in a case of h 2 /h 1 = 5, η 0 /h 1 = 1.44.In (a) and (b), the colorbar represents the horizontal velocities of the same ISW, they were calculated by the WNL and SNL theories, respectively.The quivers and black solid lines are the experimentally measured velocities and waveforms.(c) The divergence of horizontal velocities at the surface.The red line and blue line are calculated by the WNL and SNL theories, respectively.The diamonds are experimental results.The dashed lines indicate the positions where the divergences reach peaks.(d) The variation of the ratio of D p−p to wavelength with amplitude.The green dots are measured in experiments, and the red and blue dots are calculated by the WNL and SNL theories, respectively.Each color from light to dark corresponds to h 2 /h 1 from 3 to 10.

Figure 3 .
Figure 3. Panels (a), (b), and (c) are the relationships between D p−p and phase speed, wavelength, and amplitude with h 2 /h 1 = 5, respectively.The red, magenta, yellow, green, light blue, and dark blue lines represent the KdV, JKKD, BO, eKdV, MCC, and DJL equations respectively.The black diamonds and circles represent experimental results for h 1 = 0.04 and 0.08 m, respectively.Panels (d) and (e) show how the amplitude relates to the maximum values of horizontal velocity and its divergence.The lines with different colors represent the results of the DJL equation with h 2 /h 1 = 3-10, respectively.The diamonds and circles represent experimental results.

Figure 4 .
Figure 4. (a) The dark blue, light blue, green, orange, and red lines are DJL solutions with depths of 200, 600, 953, 2,762, and 3,745 m, respectively.Dashed lines of corresponding colors indicate the maximum observed ISW amplitudes η 0 at each location.(b) DJL solutions and observations at the depth of 200 m.The dark blue, gray dashed, and dot-dashed lines represent DJL solutions calculated with upper layer depths h 1 = 48 m (the average value), 30 m, and 60 m, respectively.The dots represent the observed ISWs, and their colors indicate h 1 before the arrival of ISWs.The x-axis is set to D p−p /c due to the absence of the phase speed c. (c) DJL solutions show the relationship between D p−p , maximum surface velocities, stratification, and ISW amplitudes within the same depth.The colors of the lines and the background represent h 1 and η 0 , respectively.(d) Observed η 0 and η 0 retrieved using (c).The colorbar of dots in (b), (d), and lines in (c) are the same.The red squares in (b) and (d) indicate the deformation of ISWs.