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On a model for internal waves in rotating fluids

A. Durán

A. Durán

Applied Mathematics Department, University of ValladolidValladolid, Spain
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Durán, A.
  
Dec 31, 2018
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Published Online: Dec 31, 2018
Page range: 627 - 648
Received: Oct 17, 2018
Accepted: Dec 31, 2018
DOI: https://doi.org/10.2478/AMNS.2018.2.00048
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© 2018 A. Durán, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Fig. 1

Idealized model of internal wave propagation in a two-layer interface. ρ2 > ρ1; d2 > d1; ζ(x, t) denotes the downward vertical displacement of the interface from its level of rest at (x, t).
Idealized model of internal wave propagation in a two-layer interface. ρ2 > ρ1; d2 > d1; ζ(x, t) denotes the downward vertical displacement of the interface from its level of rest at (x, t).

Fig. 2

Amplitude (a) and wavelength (λ) of a wave.
Amplitude (a) and wavelength (λ) of a wave.

Fig. 3

Numerical approximation with α = 0, β = γ = δ = 1. Computed solitary wave profiles.
Numerical approximation with α = 0, β = γ = δ = 1. Computed solitary wave profiles.

Fig. 4

Numerical approximation with α = 0, β = γ = δ = 1. Phase portraits of the computed solitary wave profiles.
Numerical approximation with α = 0, β = γ = δ = 1. Phase portraits of the computed solitary wave profiles.

Fig. 5

Numerical approximation with α = 0, β = γ = δ = 1. Speed-amplitude relations.
Numerical approximation with α = 0, β = γ = δ = 1. Speed-amplitude relations.

Fig. 6

Two-pulse for α = 0, β = −1, γ = δ = 1, p = 2, cs = 1.1 and a negative hyperbolic-secant profile as initial data for the iteration (22), (23).
Two-pulse for α = 0, β = −1, γ = δ = 1, p = 2, cs = 1.1 and a negative hyperbolic-secant profile as initial data for the iteration (22), (23).

Fig. 7

ω(k)/k vs k. Case β < 0.
ω(k)/k vs k. Case β < 0.

Fig. 8

ω(k)/k vs k for β > 0. (a) Solid line: A > 0, B > 0 (α = γ = 1/2, β = 2, δ = 1); dashed line: A < 0, B > 0 (α = γ = 1/2, β = 2, δ = 1/4); (b) Solid line: A > 0, B < 0 (α = γ = 1/2, β = δ = 1); dashed line: A < 0, B < 0 (α = 1/2, γ = 5, β = 1, δ = 1/8); (c) Magnification of (a); (d) Magnification of (b)
ω(k)/k vs k for β > 0. (a) Solid line: A > 0, B > 0 (α = γ = 1/2, β = 2, δ = 1); dashed line: A < 0, B > 0 (α = γ = 1/2, β = 2, δ = 1/4); (b) Solid line: A > 0, B < 0 (α = γ = 1/2, β = δ = 1); dashed line: A < 0, B < 0 (α = 1/2, γ = 5, β = 1, δ = 1/8); (c) Magnification of (a); (d) Magnification of (b)

Fig. 9

Amplitude vs β.
Amplitude vs β.

Fig. 10

c∗=12−β1+β4δ+(4δ−β)γδ+β4δ2$\begin{array}{} \displaystyle c^{*} = \frac{1}{2}\left(-\beta\left(1+\frac{\beta}{4\delta}\right)+(4\delta-\beta)\sqrt{\frac{\gamma}{\delta}+\left(\frac{\beta}{4\delta}\right)^{2}}\right) \end{array}$ vs β with α = 0, γ = δ = 1.
c∗=12−β1+β4δ+(4δ−β)γδ+β4δ2$\begin{array}{} \displaystyle c^{*} = \frac{1}{2}\left(-\beta\left(1+\frac{\beta}{4\delta}\right)+(4\delta-\beta)\sqrt{\frac{\gamma}{\delta}+\left(\frac{\beta}{4\delta}\right)^{2}}\right) \end{array}$ vs β with α = 0, γ = δ = 1.

Fig. 11

RMBenjamin vs Ostrovsky equations. Computed solitary wave profiles with (a) cs = 0.1, (b) cs = 0.9.
RMBenjamin vs Ostrovsky equations. Computed solitary wave profiles with (a) cs = 0.1, (b) cs = 0.9.

Fig. 12

RMBenjamin vs Ostrovsky equations. Speed-amplitude relations.
RMBenjamin vs Ostrovsky equations. Speed-amplitude relations.
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