INTRODUCTION

Intensive semi-diurnal internal tides are well-known on the Northwestern Australian Shelf [1–3], as well as in many other areas of the ocean [4]: the shelf of New England [5], the South China Sea [6], and others. On the Northwestern Australian Shelf, waves generated above the continental slope propagate towards the shore and rapidly dissipated in the shelf break area [7]. It is observed that a long internal tide becomes steeper due to nonlinear transformation, forming an internal bore and pockets of short soliton-like internal waves [8–10]. However, the previously published data provide only limited information about the pattern of the observed waves.

As his manuscript was being prepared for publication, Peter Holloway passed away suddenly due to a tragic accident and the collected material remained unpublished. The issue of highly nonlinear internal waves on the shelf nonetheless remains a focus of attention of oceanologists and the results obtained will be in demand by specialists. Improved measurement techniques allowed us to obtain time series of properties of internal waves with high temporal and spatial resolution from the continental slope–shelf transition area. The data obtained provide detailed information about the configuration of an internal wave and the changes in the wave shape, as the waves propagate, as well as the formation of an internal solitary from internal bores.

OBSERVATIONS

Three moored stations including 19 current meters were installed in a line, oriented perpendicular to isolines of the bathymetry of the Northwestern Australian Shelf. The moored stations “slope” and “break” were located at a distance of 5070 m from each other, and “break” and “shelf” ones, at a distance of 1500 m. The most distant station from the shore “slope” was located at the point with a depth of 110 m, and “break” and “shelf” were at depths of about 80 m (Fig. 1). In general, Steedman acoustic current meters were used to measure 2-min average vector values. In addition, we used the S4 Inter Ocean current meters with a 1-min sample rate and averaging of 6 min. Measurements were performed over 67 days from February 9 to April 24, 1992, with a two-day break (March 14–16, 1992), when devices were retrieved, maintained, and rearranged.

Fig. 1.
figure 1

The map of the Northwestern Australian Shelf, indicating sites of moored current meters of “slope,” “break,” and “shelf.” The left (left bottom) shows the position of moored current meters and the position of current meters.

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One of the goals of the measurement program was to study the interaction of internal wave currents with the boundary layer of the ocean bottom (see [11]). For this reason, hydrometric current meters were installed at heights of 2.5, 5.0, 10.0, and 20.0 m above the sea bottom, as well as devices higher in the water column. Due to the high sampling rate, good vertical resolution, and close proximity of the moored stations, the data obtained are well suited to detail the properties of nonlinear internal waves and observations over the transformation of waves as they propagate.

The construction of time series of current or temperature components shows a great variety of oscillations with periods ranging from semidiurnal tides to short-period (about 10 min) oscillations. Waveforms also vary widely. First, we made an attempt to arrange the observations according to the number of different types of waveforms. In terms of flow and vertical displacement, the vertical distribution of the internal wave amplitude follows a modal function. Therefore, it is reasonable to consider observations made only at the same level to study a waveform. It is observed that waves propagate towards the shore and, on average, the wave fronts are approximately aligned parallel to the bathymetric isolines. Consequently, the off-shore-to-on-shore component of the current (the coast is located at 158° east of the north, corresponding to the line passing through three moored stations) is considered to be representative both for the waveform and records from three stations at a height of 10 m above the sea bottom.

FREQUENCY SPECTRUM OF INTERNAL WAVES

In order to analyze a series of currents and to calculate the frequency spectrum, we selected the meter records made at a level of 20 m above the bottom at the “slope” point and in the direction perpendicular to the shoreline. First, the barotropic component of the tide was excluded from the total signal by removing the depth-averaged value of the current. As a result, a series of barotropic current indicators was obtained. The frequency spectrum shown in Fig. 2 was calculated based on the obtained series of the current component directed normal to the shoreline. The largest peak M2, associated with tidal regular semidiurnal waves, stands out noticeably on the descending spectrum. There is also a peak at a frequency close to the local inertial frequency f. The peak N is distinguished in the high-frequency range at frequencies close to 103 c/f associated with intensive soliton-like waves.

Fig. 2.
figure 2

Spectrum of internal waves calculated based on the data of the horizontal component of currents normal to the shoreline throughout the follow-up period.

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CLASSIFICATIONS OF WAVEFORMS

In total, 130 semidiurnal oscillations are accounted for based on the current records and from the vertical displacements calculated from the temperature time series. It became evident that these 130 waves could be divided into five different types of waveforms. In this approach, only the semidiurnal waveform is considered, and the presence or absence of short-period internal soliton-like waves does not affect the classification. Figure 3 exemplifies five types of waveforms distinguished based on the current observations. A brief description of each type is given. The number of occurrences of each waveform recorded at each of three stations is shown as a histogram in Fig. 3 (right).

Fig. 3.
figure 3

Five main characteristic waveforms observed at moored current meters. The left column shows the schematic drawings of waveforms; the middle column, examples of real waves of the corresponding type (records of the shore-normal components); the right column, the histogram distribution of wave types at three moored current meters.

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The waveform of type 1 is a bore on the frond side of the wave, superimposed on an approximately stable background flow, often associated with large-amplitude solitary waves. The waveform of type 2 is a bore on the back side of the wave; the wave of type 3 called a “square” wave is the simultaneous presence of bores on both the front and back sides of the wave. All of these waveforms are greatly nonlinear. The wave of type 4 has a small-amplitude, approximately sinusoidal oscillation, although this signal often contains multiple high-frequency oscillations. The wave of type 5 is a sinusoidal large-amplitude wave without signs of nonlinear distortion.

Waves with a bore on the back side are mainly recorded in the deep “slope” point decreasing in number as they shift to the shelf. Square waveforms (type 3) are less common. They increase in number at displacement from the “slope” to shallow stations. The number of low-amplitude sinusoidal waves largely exceeded the high-amplitude sinusoidal waves. Their number is comparable with the number of waves of type 1, propagating without change through all three stations. It should be noted that waves of types 1 and 4 are the most frequently recorded.

“Square” waves of type 3 are less common, but there is a persistent increase in the number of these waves towards shallower water stations, i.e., from “slope” to “shelf.” At the same time, there is a stable decrease in the number of waves with a bore on the back side of the type-2 wave when they propagate to the shoal from the “slope” to the “shelf.” It should be noted that strongly nonlinear waveforms (types 1– 3) are much more prevalent at all the stations than sinusoidal ones (types 4 and 5).

WAVE TRANSFORMATIONS

This section provides information on the change or transformation of waveforms between stations. When considering observations from three stations, it was established that a large number of combinations of waveform transformations are possible and the distribution statistics of these various transformations was plotted (Fig. 4).

Fig. 4.
figure 4

Various wave transformations recorded at moored current meters when the wave propagates from the continental slope to the shelf and the histogram of their distribution.

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Figure 4 shows the distribution histogram of the occurrence of all the main waveform transformations that were recorded more than twice. Frequently, one can see that the most common waveform with a bore on the front side of the wave propagates through the moored stations without changing the type. The next most common transformation is a small-amplitude wave forming a bore on the front side and then a large-amplitude wave forming a bore on the front side. Other combinations occur with approximately the same frequency.

The main features are as follows. The most common waveform is a bore on the front side of the wave (type 1), and this waveform very rarely changes between the three stations, although there are usually changes in the bore intensity. However, a large number of bores are also transformed into low-amplitude sinusoidal waves (type 4). The low-amplitude waves also often have bores on the front side. “Square” waveforms are relatively short-term. There is a single record, which shows this type of wave propagation through all three stations. This suggests that the square waveform may be an “intermediate” waveform in the transformation process. Relatively few bores are observed on the back side of the wave, and they are usually short-term.

When considering all types of wave transformations, one can see that between the “slope” and “break” stations, 62% of waves experience at least one type of transformation, while between the “break” and “shelf” stations, 28% of waves are transformed. Only 41% of waves do not change the waveform type between these three stages. A large percentage of waves that change their shape between the “slope” station and that at the shelf break indicate the strong influence of shallow water bathymetry on the transformation process when water depth varies from 109 to 78 m.

Based on the transformations observed, it is possible to propose a sequence of waveform transformation steps on their way to wave breaking. The absence of some steps in some records may be an indirect result of the fact that some steps are short-term. The suggested sequence (4–5–1–3–5) means that a low-amplitude sinusoidal wave grows in amplitude as it increases in steepness to break a large amplitude wave. Then, a bore forms on the front side of the wave, which is often associated with a series of solitons. After further steepening, a second bore forms on the back side of the wave, creating a pedestal-like waveform. Strong attenuation or breaking of the wave then quickly turns this waveform back into a low amplitude sinusoidal wave.

An alternative sequence is also possible when a wave of type 2, formed by a bore on the back side of the wave, occurs at the beginning of the sequence, as evidenced from the observation that most waves of type 2 are generated at a deeper observation point on the slope. A possible sequence, 4–5–2–3–1, is almost an inverse sequence in comparison with that described above. In this case, the small sinusoidal wave amplitude increases, and then a bore is formed on the back side of the wave. Then, a “square” wave is formed, which eventually transforms into a bore on the front side of the wave.

Each waveform can be explained as a manifestation of a bore-like shape with bores on both the front and back sides of waves. It is possible that each of the nonlinear waves represents a transformation stage between front and back bores. Low-amplitude linear waves (type 4) may be the result of a nonlinear wave breaking. A large-amplitude linear wave (type 5) was presumably subject to a strong transformation in a shoal after further propagation.

Let us emphasize that within the framework of this work it was established that not only the traditional internal bores, but also unusual “square” waves are generated from the transformation of the internal tides propagating along the shelf. Despite these waves representing an intermediate phase of nonlinear transformation, relatively short-term, this phase continues to exist during for at least several hours. These waves are generated when a baroclinic tide moves from deep water to a shoal. Propagating from the slope to the shelf across the shelf break, a tidal wave with a bore on the back side becomes steeper frontally. Here, the back slope steepness remains. As a result, a square wave is generated (Fig. 5a). According to our estimates, a square wave propagates about 6.5 km over four hours at a velocity of about 0.45 m/s. A rectangular wave phenomenon can be explained by applying the model presented in [12].

Fig. 5.
figure 5

(a) Conversion of a tidal wave with a bore on the retreating side into a square wave (in red) when approaching the shallow-water coast; (b) conversion of a square wave (in gray) into a wave with a bore on the retreating side when approaching the shallow-water coast.

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Upon entering a shoal, the leading edge of a wave enters the zone where the thermocline is located close to the seafloor. This intensifies the nonlinearity coefficient and leads to steepening of the leading edge.

The case of transforming a square wave into a wave with a steep bore on the back side, which is also possible when entering the shoal, is also of interest (Fig. 5b).

The occurrence of bores depends on the conditions of the water environment where the internal tides propagate. In particular, when applying the Korteweg–De Vries equation, it depends on the nonlinearity coefficient value α(n). When a thermocline is located close to the sea surface, the value α is positive. In this case, the bores appear on the back side of the wave (type 3). When the thermocline is close to the seafloor, the value α is negative and bores appear on the front side of the wave (type 1). The background current also determines the change of sign of α. Work [12] presents a model based on an extended modified Korteweg–De Vries equation developed for sinusoidal internal tidal waves that nonlinearly transform as they propagate across the shelf and generate soliton-like wave trains. The combined effect of nonlinearity and rotation leads to the occurrence of a number of interesting waveform features, including solitons of the polarities, “thick” solitons, and sharp waves with steep fronts. Rotation increases the phase velocity of the long internal tide, reduces the number of internal soliton waves generated from the long wave, and changes the wave shape.

The results of the numerical simulation are compared with current and temperature observations over the internal wave field on the Northwestern Australian Shelf, which is confirmed by many features of the generalized KdV model.

CONCLUSIONS

This paper presents an analysis of the longitudinal record of currents and temperature variations made in a section of the Northwestern Australian Shelf, a known location of large nonlinear internal tides. The nonconventional analysis of 130 semidiurnal tidal waves allowed us to reveal their diversity and to distinguish the most common phases of their nonlinear transformations. As a result, five characteristic types of tidal semidiurnal waves observed on the Australian shelf were identified. They are listed below in order of frequency of occurrence, from the highest to the lowest. The wave of type 1 is the most common type of nonlinear wave with a bore on the front side of the wave. The wave of type 4, the second most common, is a small-amplitude, approximately sinusoidal wave. Its origin can be attributed to the breaking of strongly nonlinear internal tidal waves propagating along the shelf. The third most common wave is type 2, a nonlinear wave with a bore on the back side of the wave. The wave of type 3, called a “square” wave, is less common. This wave is characterized by the simultaneous presence of bores on both the front and back sides of the wave. The wave of type 5, a sinusoidal wave with no signs of nonlinear distortions, occurs least often.

Regarding nonlinear transformations, it should be noted that only 11 types of transformations of nonlinear internal tidal waves propagating along the shelf were revealed during observations. Let us note the most frequent cases. The greatest number of cases is associated with the propagation of nonlinear waves of type 1 without a change in shape. There are almost two times fewer cases when a nonlinear wave with a bore on the front side, after propagating by two stations, is destroyed on the approach to the third station being transformed into a small-amplitude sinusoidal wave. There are common cases when the amplitude of a small-amplitude wave increases on its way from the first point of observation (slope) and turns into a nonlinear wave with a bore on the front side. Then, this waveform propagates along the rest of the section route. The propagation of a small-amplitude wave along the route is also a common occurrence. However, apparently, in the water area closer to the shore, the following effects are still observed: an accumulation of nonlinear effects with increasing amplitude, steepening of the front side, and further breaking of the wave. It is also interesting to observe a square wave propagating by the “slope” station with subsequent breaking. The cases of transformation of a wave with a bore on the back side, propagating by the “slope” station into a highly nonlinear wave with a bore on the front side are also of interest.

The present paper concentrates on transformations of internal tides, although short-period waves (soliton-like waves) occur very often and are worth a more careful look as an important element of the nonlinear transformation process. It should be noted that, while analyzing the records available, we have recognized that short waves have signs indicating their belonging to the second mode. However, this has not become an issue of discussion. A recently published paper devoted the observation of internal waves in the deeper part of the Australian Northwestern Shelf presents data on the recorded thermistor chains of internal wave solitons of the second mode [13].