Coupled triads in the dynamics of internal waves: Case study using a linearly stratified fluid

In a linearly stratified fluid, two sets of resonant triads with one member in common, hence coupled, can arise for a wide range of wave numbers where all five waves travel in the same direction. The nonlinear dynamics is investigated by deriving the evolution equations of slowly varying wave packets. The dependence of the interaction coefficients on the channel depth and the buoyancy frequency are explicitly demonstrated. Coupling may produce instabilities with growth rates bigger than those displayed by either triad in isolation. Similar properties are expected for layered fluids with shear currents. By drawing on a recent link between modulation instability and the occurrence of rogue waves, surprisingly large displacements in the interior of the fluid are thus more likely. For applications to oceanography, the interaction coefficients of the evolution equations will affect the magnitude of the actual amplitude of an internal rogue wave. Computations indicate that displacements much larger than those of surface rogue waves are possible. From a more theoretical perspective of wave resonance through the Madelung representation, special ranges of dynamical phase angles are identified for the existence of time invariants. Finally, numerical simulations of a simplified system are performed to highlight the spontaneous generation of modes due to coupling.

Figure 1

The dispersion relation, angular frequency ω versus wave number $k$, for a fluid of constant buoyancy frequency $N_0$ with $m$ being the mode number ($N_0=1, H=1$).

Figure 2

Graphical construction of a triad.

Figure 3

Schematic diagram of coupled triads [Eqs. (1) and (5)] with all five participating waves traveling in the same direction. A solid dot on the $m=3$ branch represents the common member, with modes 1, 5, 3 forming one triad and modes 2, 4, 3 the other triad.

Figure 4

Computations of the triad resonance condition, Eq. (7), for typical input parameters ($k_1=0.5$) showing the existence of two possible values for $k_5$. We adopt the smaller roots ($k_5=2.8787$) in subsequent calculations.

Figure 5

Variation of the interaction coefficients with the channel depth $H$.

Figure 6

Direct proportional relation between the interaction coefficients $r$ and the buoyancy frequency $N$ for various fluid scenarios. For each value of $N$, the configuration under investigation is a stratified fluid with constant buoyancy frequency $N=N_0$.

Figure 7

Both the (1, 5, 3) and (2, 4, 3) triads are unstable by themselves, but coupling produces an even stronger instability.

Figure 8

Triads on their own exhibit passband instability only. Top: isolated (1, 3, 5) triad; Middle: isolated (2, 3, 4) triad. Bottom: Coupling induces baseband (small wave number) instability (solid curve) and additional passband instability (dotted curve).

Figure 9

Rogue waves for the three components an isolated triad (intensities $|{\mathrm{\Psi }}_n{|}^2, n=1,2,3$ vs space and time). Left column: Profile with one peak (${\gamma }_3=0$). Right column: Profile with two peaks (${\gamma }_3=1$); ${\gamma }_1={\gamma }_2=q=1, V_1=2, V_2=1$.

Figure 10

Scaling factor for converting the dimensional triad system [Eq. (13a)] to a nondimensional one like Eq. (19), implying that the magnitude of an internal rogue wave can attain big amplitude due to the large scaling factor.

Figure 11

The common member of the two coupled triads excites all the participating modes. Modes classified as “present (absent)” initially will be given amplitude 1.0/0.0001 at time $\tau =0$. Only the common member $A_3$ with wave number $k_3$ is present initially. The normalized interaction coefficients are given by $s_1=+1, s_2=-1, s_{3a}=-1, s_{3b}=+1, s_4=-1, s_5=+1$.

Figure 12

(a) Energy transfer arises from the coupling of two triads through the common member. Initially only $A_1, A_5$ have finite amplitude with $|A_1|=1, |A_2|=0.0001, |A_3|=0.01, |A_4|=0.0001, |A_5|=1$. A magnified view for $b_3$ is inserted on the right to show the numerically small (but nonzero) initial condition. Energy flows from the $k_3=k_1+k_5$ triad to the $k_3=k_2+k_4$ triad through the common member $A_3$ with wave number $k_3$. (b) The temporal evolution of the dynamical phase factors ${\theta }_1,{\theta }_2$: While each phase oscillates sinusoidally, the difference ${\theta }_1–{\theta }_2$ approaches asymptotically the value of $-\pi$ for large time.

Figure 13

Effect of the dynamical phase factors [Eq. (22)] in the evolution of the wave profiles. Top: ${\theta }_1=0, {\theta }_2=\pi /2$; Bottom: ${\theta }_1=\pi /2, {\theta }_2=0$. Initial conditions are $b_1=b_5=1, b_2=b_4=0.0001, b_3=0.001$.

Figure 14

(a) Effect of varying interaction coefficients; values as given by Eq. (30), except that $s_{3b}=3$. (b) Same as (a) except that $s_{3b}=15$.

Figure 15

An example of partial or minimal transfer of energy, as only modes of one triad $A_2, A_3, A_4$ are excited. The interaction coefficients have sign patterns as $s_1=+1, s_2=-1, s_{3a}=-1, s_{3b}=+1, s_4=-1, s_5=-1$.

Figure 16

(a) Schematic diagram of a two-layer fluid with shear flow in the upper layer. (b) Dispersion relation (angular frequency ω vs wave number $k$) for surface and interfacial waves for a moderate linear shear current $Fz, F=0.4$. (c) “Close-up (zoom-in)” view of the region where the dispersion branches approach each other. (d) Schematic diagram of new coupled triad resonances by drawing the dispersion branches, “broken line” for the backward-moving interfacial mode and “dotted line” for the backward-moving surface mode, with a mode represented by a small circle as the common member of the coupled triads. Two distinct long waves (modes with small wave numbers) can participate in these resonances.