Some features of solitonic waves propagating in intermediate waters

This work addresses the problem of weakly non linear ocean waves propagation in intermediate waters. We have shown that the propagation of waves similar to hole solitons, as well as progressive waves of high intensity multipeaks are likely to occur in this non linear regime. In addition, we note that along intermediate waters, these particular non-linear waves satisfy a wave equation model similar to Korteweg de Vries equation, and their propagation features strongly depend on the initial conditions adopted to the present problem.

Key words
ocean waves; extreme waves; non linear waves; solitons in fluids; hydrodynamics

Authorship

LUIZ GALLISA GUIMARÃES

Universidade Federal do Rio de Janeiro-UFRJ, Programa de Engenharia Oceânica-COPPE, Centro de Tecnologia da UFRJ, Av. Athos da Silveira Ramos, Cidade Universitária, 21941-611 Rio de Janeiro, RJ, BrazilUniversidade Federal do Rio de Janeiro-UFRJBrazilRio de Janeiro, RJ, BrazilUniversidade Federal do Rio de Janeiro-UFRJ, Programa de Engenharia Oceânica-COPPE, Centro de Tecnologia da UFRJ, Av. Athos da Silveira Ramos, Cidade Universitária, 21941-611 Rio de Janeiro, RJ, Brazil

https://orcid.org/0000-0001-5699-1251

E-mail: LuLa@if.ufrj.br; LGG@oceanica.ufrj.br

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables

Figures (10)
Tables (2)

Thumbnail

Figure 1
for a wave with period

𝐓 = 5 𝐬

, the panel a) shows the behavior of the wavelengths

𝛌

and

𝛌_{0}

as a function of the depth

𝐝

and the deep water wavelength

𝛌_{\infty}

. It is also noted that

𝛌_{0} = 𝛌_{\infty}

at depth

𝐝_{𝐢}

. The panel b) shows that in shallow waters

𝛌 / 𝐝_{𝐢} \approx 5

is limited between

𝛅_{0}^{- 1}

and

𝛅_{\infty}^{- 1}

, these results are in accordance with those pioneering discussed in (Lighthill 1978).

Thumbnail

Figure 2
shows in panel a) the wave refraction angle

𝛉

as a function of the dimensionless depth

𝐡

. The panel b) shows the behavior of the phase

𝐜

and group

𝐮

velocities as

𝐡

varies. Note that

𝐮

reaches a maximum value

𝐮_{𝐦 𝐚 𝐱}

𝐡_{𝐢} \equiv 1

, and that around these intermediate waters (

𝐝 \approx 𝐝_{𝐢}

) the wave refractive and dispersive effects are still very significant.

Thumbnail

Figure 3
sketches the behavior of the CCCDF, in the cases associated with refraction and dispersion of waves as well as for the situation where the refraction and dispersion effects can occur simultaneously. Note that the panels a) and b) show respectively how in these cases the Bersntein’s polynomial

𝐁

and its derivative

𝐁^{'}

behave as

𝐡

varies.

Thumbnail

Figure 4
compares in panels a) and b) the expansions in Chebychev polynomials

𝐏_{𝐓}

and

𝐐_{𝐓}

for the functions

𝐏

and

𝐐

respectively. Note in a) that

𝐏_{𝐓}

fits

𝐏

well in the whole range

𝐡_{𝐬} \leq 𝐡 \leq 𝐡_{𝐝}

, while the panel b) shows that

𝐐_{𝐓}

also fits

𝐐

well for depths in deep and intermediate waters, but the difference between

𝐐_{𝐓}

and

𝐐

increases around shallow waters. In both panels, the full dots delimit the range between deep and shallow waters

𝐡_{𝐬} \leq 𝐡 \leq 𝐡_{𝐝}

Thumbnail

Figure 5
illustrates the propagation behavior of the

𝚫 𝛀

in the plane

(𝛏, 𝛕)

. Both graphs, the first showing the scattered points in the panel (a) and the second related to the level surfaces represented in the panel (b), show that in deep waters (

𝛏 \approx 0

) the waveform

𝚫 𝛀

is a deeper trough (see color scale in bottom panel–b), and it becomes a hole soliton in intermediate waters (

𝛏 \approx 1

), but it cannot propagate in shallow waters as

𝛏 \to \infty

Thumbnail

Figure 6
shows a grayscale map for the

𝚫 𝛀

normalized waveform behavior as the dimensionless time

𝛕

and depth

𝐡

vary. In this map the widest and deepest water vacancies are represented by the darkest shades and they occur in large scale in deep waters (

𝐡 ≫ 𝐡_{𝐝} \approx 2.1

). It is also noted that in intermediate waters (

𝐡 \approx 1

), the hole soliton shape is well defined, and as this waveform reaches shallow water it becomes narrower and it disappears as

𝐡 ≪ 𝐡_{𝐬} \approx 0.28

Thumbnail

Figure 7
for

𝐒 = - 1

in Eqs. (65 and 66) shows some features of

𝚫 𝛀

propagation in the plane

(𝛏, 𝛕)

. It is observed both in the panel (a) in the form of scattered points, and in the level surface of the panel (b) that there are patterns of intense peaks regularly spaced. It is also noted that for

𝛏 \approx 1

there is a rather extensive barrier of these intense peaks around these intermediate waters.

Thumbnail

Figure 8
for the case where

𝐒 = - 1

in Eqs. (65 and 66), shows in a grayscale map the temporal evolution of

𝚫 𝛀

propagation at depths in the range

𝐡_{𝐬} \leq 𝐡 \leq 𝐡_{𝐝}

. Since in this figure the brightest tones represent intense amplitudes, it can be observed that around depths

𝐡 = 𝐡_{𝐜, - 1}

(see Eq. (67)) a barrier of high wave amplitudes is established, moreover it can also be observed that the pattern of these intense peaks are quite different in shallow waters from those in deeper waters.

Thumbnail

Figure 9
for

𝐒 = 11

in Eqs. ((65) and (66)), shows in a gray scale map the behavior of the

𝚫 𝛀

temporal evolution as a function of

𝐡

. In this map the brightest tones represent the greatest amplitude values of the wave. It is noted that a barrier for these oscillating waves with high amplitudes occurs at depth

𝐡 = 𝐡_{𝛋, 11}

(see Eq. (68)), and that after evanescently crosses this barrier, the wave becomes a hole soliton like. Moreover, this hole soliton reaches another barrier at

𝐡 = 𝐡_{𝐜, 11}

(see Eq. (69)) which forbids its propagation in shallow water.

Thumbnail

Figure 7
for

𝐒 = - 1

in Eqs. (65 and 66) shows some features of

𝚫 𝛀

propagation in the plane

(𝛏, 𝛕)

. It is observed both in the panel (a) in the form of scattered points, and in the level surface of the panel (b) that there are patterns of intense peaks regularly spaced. It is also noted that for

𝛏 \approx 1

there is a rather extensive barrier of these intense peaks around these intermediate waters.

Thumbnail

Figure 8
for the case where

𝐒 = - 1

in Eqs. (65 and 66), shows in a grayscale map the temporal evolution of

𝚫 𝛀

propagation at depths in the range

𝐡_{𝐬} \leq 𝐡 \leq 𝐡_{𝐝}

. Since in this figure the brightest tones represent intense amplitudes, it can be observed that around depths

𝐡 = 𝐡_{𝐜, - 1}

Thumbnail

Figure 9
for

𝐒 = 11

in Eqs. ((65) and (66)), shows in a gray scale map the behavior of the

𝚫 𝛀

temporal evolution as a function of

𝐡

. In this map the brightest tones represent the greatest amplitude values of the wave. It is noted that a barrier for these oscillating waves with high amplitudes occurs at depth

𝐡 = 𝐡_{𝛋, 11}

(see Eq. (68)), and that after evanescently crosses this barrier, the wave becomes a hole soliton like. Moreover, this hole soliton reaches another barrier at

𝐡 = 𝐡_{𝐜, 11}

(see Eq. (69)) which forbids its propagation in shallow water.

Thumbnail

Figure A1
Shows for some wave periods

𝐓

the behavior of

sin 𝛉 (𝐡)

as a function of the normalized depth

𝐡

. For an accuracy of

10^{- 6}

, the values of

sin (𝛉)

were calculated using the algorithm(70) to solve numerically Eq. (15). Notice that the wavelength

𝛌 (𝐓, 𝐝)

satisfies the Snell’s law(14),

𝛌 = 𝛌_{\infty} sin (𝛉)

Thumbnail

Table I
based on CCCDF summarizes the limits between the regions here established as normalized shallow depth

𝐡_{𝐬} \approx 0.275

, intermediate depth at

𝐡_{𝐢} = 1

and deep waters a part from

𝐡_{𝐝} \approx 2.11

respectively. Besides, it is also shown the ratio

𝛅 = 𝐝 / 𝛌

and the refraction angle

𝛉

related to these waters.

Thumbnail

Table II
shows the numerical values of the coefficients

𝐋_{𝐧}

and

𝐍_{𝐧}

related to the Chebyshev polynomials

𝐏_{𝐓}

and

𝐐_{𝐓}

expansions respectively.

Figure 1 for a wave with period $𝐓 = 5 𝐬$ , the panel a) shows the behavior of the wavelengths $𝛌$ and $𝛌_{0}$ as a function of the depth $𝐝$ and the deep water wavelength $𝛌_{\infty}$ . It is also noted that $𝛌_{0} = 𝛌_{\infty}$ at depth $𝐝_{𝐢}$ . The panel b) shows that in shallow waters $𝛌 / 𝐝_{𝐢} \approx 5$ is limited between $𝛅_{0}^{- 1}$ and $𝛅_{\infty}^{- 1}$ , these results are in accordance with those pioneering discussed in (Lighthill 1978).

Figure 2 shows in panel a) the wave refraction angle $𝛉$ as a function of the dimensionless depth $𝐡$ . The panel b) shows the behavior of the phase $𝐜$ and group $𝐮$ velocities as $𝐡$ varies. Note that $𝐮$ reaches a maximum value $𝐮_{𝐦 𝐚 𝐱}$ at $𝐡_{𝐢} \equiv 1$ , and that around these intermediate waters ( $𝐝 \approx 𝐝_{𝐢}$ ) the wave refractive and dispersive effects are still very significant.

Figure 3 sketches the behavior of the CCCDF, in the cases associated with refraction and dispersion of waves as well as for the situation where the refraction and dispersion effects can occur simultaneously. Note that the panels a) and b) show respectively how in these cases the Bersntein’s polynomial $𝐁$ and its derivative $𝐁^{'}$ behave as $𝐡$ varies.

Figure 4 compares in panels a) and b) the expansions in Chebychev polynomials $𝐏_{𝐓}$ and $𝐐_{𝐓}$ for the functions $𝐏$ and $𝐐$ respectively. Note in a) that $𝐏_{𝐓}$ fits $𝐏$ well in the whole range $𝐡_{𝐬} \leq 𝐡 \leq 𝐡_{𝐝}$ , while the panel b) shows that $𝐐_{𝐓}$ also fits $𝐐$ well for depths in deep and intermediate waters, but the difference between $𝐐_{𝐓}$ and $𝐐$ increases around shallow waters. In both panels, the full dots delimit the range between deep and shallow waters $𝐡_{𝐬} \leq 𝐡 \leq 𝐡_{𝐝}$ .

Figure 5 illustrates the propagation behavior of the $𝚫 𝛀$ in the plane $(𝛏, 𝛕)$ . Both graphs, the first showing the scattered points in the panel (a) and the second related to the level surfaces represented in the panel (b), show that in deep waters ( $𝛏 \approx 0$ ) the waveform $𝚫 𝛀$ is a deeper trough (see color scale in bottom panel–b), and it becomes a hole soliton in intermediate waters ( $𝛏 \approx 1$ ), but it cannot propagate in shallow waters as $𝛏 \to \infty$ .

Figure 6 shows a grayscale map for the $𝚫 𝛀$ normalized waveform behavior as the dimensionless time $𝛕$ and depth $𝐡$ vary. In this map the widest and deepest water vacancies are represented by the darkest shades and they occur in large scale in deep waters ( $𝐡 ≫ 𝐡_{𝐝} \approx 2.1$ ). It is also noted that in intermediate waters ( $𝐡 \approx 1$ ), the hole soliton shape is well defined, and as this waveform reaches shallow water it becomes narrower and it disappears as $𝐡 ≪ 𝐡_{𝐬} \approx 0.28$ .

Figure 7 for $𝐒 = - 1$ in Eqs. (65 and 66) shows some features of $𝚫 𝛀$ propagation in the plane $(𝛏, 𝛕)$ . It is observed both in the panel (a) in the form of scattered points, and in the level surface of the panel (b) that there are patterns of intense peaks regularly spaced. It is also noted that for $𝛏 \approx 1$ there is a rather extensive barrier of these intense peaks around these intermediate waters.

Figure 8 for the case where $𝐒 = - 1$ in Eqs. (65 and 66), shows in a grayscale map the temporal evolution of $𝚫 𝛀$ propagation at depths in the range $𝐡_{𝐬} \leq 𝐡 \leq 𝐡_{𝐝}$ . Since in this figure the brightest tones represent intense amplitudes, it can be observed that around depths $𝐡 = 𝐡_{𝐜, - 1}$ (see Eq. (67)) a barrier of high wave amplitudes is established, moreover it can also be observed that the pattern of these intense peaks are quite different in shallow waters from those in deeper waters.

Figure 9 for $𝐒 = 11$ in Eqs. ((65) and (66)), shows in a gray scale map the behavior of the $𝚫 𝛀$ temporal evolution as a function of $𝐡$ . In this map the brightest tones represent the greatest amplitude values of the wave. It is noted that a barrier for these oscillating waves with high amplitudes occurs at depth $𝐡 = 𝐡_{𝛋, 11}$ (see Eq. (68)), and that after evanescently crosses this barrier, the wave becomes a hole soliton like. Moreover, this hole soliton reaches another barrier at $𝐡 = 𝐡_{𝐜, 11}$ (see Eq. (69)) which forbids its propagation in shallow water.

Figure 7 for $𝐒 = - 1$ in Eqs. (65 and 66) shows some features of $𝚫 𝛀$ propagation in the plane $(𝛏, 𝛕)$ . It is observed both in the panel (a) in the form of scattered points, and in the level surface of the panel (b) that there are patterns of intense peaks regularly spaced. It is also noted that for $𝛏 \approx 1$ there is a rather extensive barrier of these intense peaks around these intermediate waters.

Figure 8 for the case where $𝐒 = - 1$ in Eqs. (65 and 66), shows in a grayscale map the temporal evolution of $𝚫 𝛀$ propagation at depths in the range $𝐡_{𝐬} \leq 𝐡 \leq 𝐡_{𝐝}$ . Since in this figure the brightest tones represent intense amplitudes, it can be observed that around depths $𝐡 = 𝐡_{𝐜, - 1}$ (see Eq. (67)) a barrier of high wave amplitudes is established, moreover it can also be observed that the pattern of these intense peaks are quite different in shallow waters from those in deeper waters.

Figure 9 for $𝐒 = 11$ in Eqs. ((65) and (66)), shows in a gray scale map the behavior of the $𝚫 𝛀$ temporal evolution as a function of $𝐡$ . In this map the brightest tones represent the greatest amplitude values of the wave. It is noted that a barrier for these oscillating waves with high amplitudes occurs at depth $𝐡 = 𝐡_{𝛋, 11}$ (see Eq. (68)), and that after evanescently crosses this barrier, the wave becomes a hole soliton like. Moreover, this hole soliton reaches another barrier at $𝐡 = 𝐡_{𝐜, 11}$ (see Eq. (69)) which forbids its propagation in shallow water.

Table I based on CCCDF summarizes the limits between the regions here established as normalized shallow depth $𝐡_{𝐬} \approx 0.275$ , intermediate depth at $𝐡_{𝐢} = 1$ and deep waters a part from $𝐡_{𝐝} \approx 2.11$ respectively. Besides, it is also shown the ratio $𝛅 = 𝐝 / 𝛌$ and the refraction angle $𝛉$ related to these waters.

$h_{*}$	$δ (h_{*})$	$θ (h_{*})$
$h_{s}$	$δ_{s} \approx 0.087$	$θ_{s} = {30.0}^{o}$
$h_{i}$	$δ_{i} \approx 0.191$	$θ_{m} \approx {56.5}^{o}$
$h_{d}$	$δ_{d} \approx 0.345$	$θ_{d} \approx {77.0}^{o}$

Table II shows the numerical values of the coefficients $𝐋_{𝐧}$ and $𝐍_{𝐧}$ related to the Chebyshev polynomials $𝐏_{𝐓}$ and $𝐐_{𝐓}$ expansions respectively.

$n$	$L_{n}$	$N_{n}$
1	22660/24111	32381/27242
3	-7741/100000	5827/98707
5	53/12500

Figure A1 Shows for some wave periods $𝐓$ the behavior of $sin 𝛉 (𝐡)$ as a function of the normalized depth $𝐡$ . For an accuracy of $10^{- 6}$ , the values of $sin (𝛉)$ were calculated using the algorithm(70) to solve numerically Eq. (15). Notice that the wavelength $𝛌 (𝐓, 𝐝)$ satisfies the Snell’s law(14), $𝛌 = 𝛌_{\infty} sin (𝛉)$ .

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Some features of solitonic waves propagating in intermediate waters

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Mensagem

[1] E-mail: LuLa@if.ufrj.br; LGG@oceanica.ufrj.br