Motivation

Internal tides are internal waves at tidal frequencies generated by the interaction of usual (barotropic) tides and topographic slopes. However, they are known to contain substantial components that cannot be predicted by (deterministic) harmonic analysis, because random oceanic variability modulates their amplitudes and phases. Although the statistical aspects of these random nonharmonic components are important basis for many purposes, they have not been considered in detail. This study develops the first detailed statistical model of nonharmonic (also called incoherent, nonstationary, or non-phase-locked) internal tides observed at a fixed location.

Theoretical model

Since it is well-known that internal tides at an observation location can consist of waves arriving from multiple and remote sources^4,5), it is expected from the central limit theorem in statistics that the process becomes Gaussian as the number of wave sources increases. As a model of nonharmonic internal tides, we consider a sinusoidal signal with a constant angular frequency $\omega$, a random amplitude $A$, and a random phase lag $\Theta$ observed at a fixed location, and assume that this signal consists of independent and non-uniformly distributed sinusoidal signals from many sources. This can be written as \begin{align} A\text{e}^{-\text{i}\Theta}\text{e}^{\text{i}\omega t} &=(X+\text{i} Y)\text{e}^{\text{i}\omega t}\\ &= \sum_{j=1}^N (X_j+\text{i} Y_j) \text{e}^{\text{i}\omega t}, \end{align} where $t$ is time, $(X+\text{i} Y)$ is the Cartesian representation of the complex-valued amplitude $A\text{e}^{-\text{i}\Theta}$, and $N$ is the number of wave components (sources). If $N$ is sufficiently large, and if $X$ and $Y$ are independent, the PDF approaches the joint Gaussian distribution. By writing it in polar coordinates, we get the amplitude PDF $f_A$ and the phase PDF $f_\Theta$: \begin{align} f_{A}(a)&\sim\dfrac{2a}{\sigma_{A}^2\sqrt{1-b^2}}\exp\left(-\dfrac{a^2}{(1-b^2)\sigma_{A}^2}\right)I_0\left(\dfrac{ba^2}{(1-b^2)\sigma_{A}^2}\right),\\ f_{\Theta}(\theta)&\sim \dfrac{1}{2\pi}\dfrac{\sqrt{1-b^2}}{1-b\cos 2(\theta-\varphi)},\\ \sigma_{A}^2&=\sigma_{X}^2+\sigma_{Y}^2,\\ b&=\sigma_{A}^{-2}|\sigma_{X}^2-\sigma_{Y}^2|. \end{align} Here, $\sigma_{X}^2$ and $\sigma_{Y}^2$ are respectively the variance of $X$ and $Y$, $\varphi$ is the mean phase lag, and $I_0$ is the modified Bessel function of the first kind of the order 0.

Observations

To test the applicability of the above statistical model, we compared the model PDFs with observed PDFs at the PIL200 location (of Australian IMOS) on the Australian North West Shelf. The observations were made at about 200-m water depth over about 2.5 years. The data were processed as follows.

1. Estimated amplitudes of the dominant internal-wave mode, called vertical mode one, at 15-min intervals (black line in Fig. 1a).
2. Calculated harmonic internal tides by applying harmonic analysis to the whole record (red line in Fig. 1a).
3. Calculated nonharmonic internal tides by subtracting the harmonic internal tides from the original amplitudes.
4. Extracted envelope amplitudes and phase lags for the diurnal (24 h), semi-diurnal (12 h), and quarter-diurnal (6h) components by band-pass filtering (Fig. 1c,d).

Results

The model PDFs were statistically not different from the observed PDFs for both envelope amplitudes and phases for the diurnal, semi-diurnal, and quarter-diurnal nonharmonic internal tides (Fig. 2).

Discussion

To my knowledge, this is the first study that developed a detailed statistical model of nonharmonic internal tides. The statistical information, such as the amplitude PDF, provides an important basis for many purposes, as seen in the example of surface waves for engineering applications.

The results of this study may appear trivial because it follows from the central limit theorem; however, this is the first study that showed the importance of viewing nonharmonic internal tides as the superposition of many wave components. For example, previous studies considered one wave-component model in which the amplitude and phase are normally distributed^1,2): \begin{align} \label{simpleModelPrevious} \eta=(r+A) \text{e}^{\text{i}(\omega t-\Theta)}.\tag{1} \end{align} Here, $\eta$ is the displacement amplitude of internal tides, and $r$ is the harmonic amplitude. However, this study suggests that the corresponding simple model in many wave-component case is \begin{align} \label{simpleModelNew} \eta=r \text{e}^{\text{i}(\omega t-\varphi_0)}+(X+\text{i} Y) \text{e}^{\text{i}(\omega t-\varphi)},\tag{2} \end{align} where $X$ and $Y$ are normally distributed. Since \eqref{simpleModelPrevious} has to assume small $A$, a broad Gaussian form from \eqref{simpleModelNew} (Fig. 3a) cannot be represented by \eqref{simpleModelPrevious} (Fig. 3b,c). The difference can be important because \eqref{simpleModelPrevious} has been used to analyse observed nonharmonic internal tides^{1-3, 6)}, but this study suggests that \eqref{simpleModelNew} is likely to be more appropriate.

Finally, this study unfortunately suggests difficulty in process investigation of nonharmonic internal tides based on their observations at one location, because their variability and PDFs tend to approach the universal form by statistical principles, regardless of the details of individual wave components. For process investigation, the proposed statistical model was used to develop a model suite for Combined Adjoint, Statistical, and Stochastic Modelling of Nonharmonic Internal Tides.

Related Publications

Details of this study are available in

Shimizu, K. 2025. Process-based modelling of nonharmonic internal tides using adjoint, statistical, and stochastic approaches - Part 1: Statistical model and analysis of observational data. Ocean Science, 21: 2233-2253, https://doi.org/10.5194/os-21-2233-2025

The data from the above study are available in

Shimizu, K. 2024. Data from: Process-based modelling of nonharmonic internal tides using adjoint, statistical, and stochastic approaches. Part I: statistical model and analysis of observational data, Zenodo, https://doi.org/10.5281/zenodo.13999868.

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