Internal waves in a shear background current: Transition from solitary-wave regime to dispersive-wave regime

Nonlinear internal waves propagating in a quasi-two-layer stratification with a shear background current are investigated by means of high-resolution direct numerical simulations. The simulations are performed in a two-dimensional rectangular domain, representing a laboratory-scale tank of finite length. The internal waves are generated using the lock-release mechanism, while the background shear currents are induced by basin-scale internal seiches created using a tilted tank suddenly returned to the untilted configuration. Both mechanisms are readily realizable in a laboratory environment. Depending on the configuration of the initial density profile, both internal solitary-like waves (ISWs) and finite-amplitude dispersive wave trains (DWTs) may be observed in the simulations. The ISWs are observed when the pycnocline is located relatively far away from the middepth and/or the background shear is oriented against the wave-induced velocity shear. Comparison of the waveforms observed in the simulations to those described by the fully nonlinear Dubreil-Jacotin–Long (DJL) theory suggests that these waves are indeed solitary-like, implying that laboratory experiments of ISWs propagating in a shear background current can be realized in a relatively straightforward manner. As the pycnocline approaches the middepth and/or the background shear reverses its polarity and increases its magnitude, the waves tend to have a smaller amplitude and a larger half-width. When a wave's amplitude is less than approximately 3% of its half-width, the dispersive effect becomes dominant and the wave loses its solitary-like form. In this case, a finite-amplitude DWT is formed instead, in which the leading wave has a reduced propagation speed from the ISW propagation speed. Analysis based on the DJL theory suggests that exact solitary waves of similar amplitude in such a background environment have an extremely large half-width that is on the same order of magnitude as the finite length of the simulation domain, implying that their formation cannot be supported in a laboratory environment.

Figure 1

Schematic diagrams of the initial density field, with the initial wave generated at the seiche's (a) crest and (b) trough.

Figure 2

Time series plot of the maximum seiche-induced horizontal velocity (solid curve) and vertical velocity (dash-dotted curve) for the case C0 from $t=10$ to 200 s.

Figure 3

Hovmöller plots of the horizontal velocity along the inviscid top boundary for the cases (a) C10, (b) I10, and (c) T10.

Figure 4

Horizontal velocity fields of (a) C10, (b) I10, and (c) T10 at $t=100$ s. The pycnoclines are indicated by the black contours and the waves' crests (determined by the locations of the maximum wave-induced horizontal velocity) are indicated by the dashed lines.

Figure 5

Visualization of Table 2, with amplitude versus (a) half-width and (b) propagation speed for the ISWs at $t=100$ s. From left to right, circles represent the cases C10, I10, and T10, triangles represent the cases C15, I15, and T15, diamonds represent the cases C20, I20, and T20, and squares represent the cases C25, I25, and T25.

Figure 6

Comparison of the horizontal velocity fields between (a) C10L at $t=200$ s and (b) C10 at $t=100$ s. The difference between (a) and (b), taken by matching the locations of the waves' crests in (a) and (b), is shown in (c). The pycnoclines are indicated by the black curves and the waves' crests are indicated by the dashed lines.

Figure 7

Comparison of the simulation result to the DJL solutions along the wave's crest, for the case C10 at $t=100$ s. (a) Maximum difference in $\rho$ normalized by $\mathrm{\Delta }\rho$. (b) Root mean square of the difference in $u$ normalized by $[max(u)-min(u\left)\right]$. Note that the APE is scaled by ${\rho }_{0}gH$ in a two-dimensional domain and is in units of ${\mathrm{m}}^2$; see Eq. (3).

Figure 8

Horizontal velocity fields in (a) the simulation C10 at $t=100$ s and (b) the DJL solution, with pycnoclines indicated by the black contours. (c) Difference in the horizontal velocity (pseudocolor) and density (black contours) between the simulation result and the DJL solution, computed by interpolating the velocity and density fields of (a) onto the grid of (b) and then taking the difference.

Figure 9

Shaded contours showing (a) solitary-wave width and (b) separation length between a solitary wave and linear waves, predicted by the Gardner equation for a two-layer fluid. In (b), the curve along which the split length equals the tank length is shown in green. In both panels, the shaded quantities are scaled by the tank's length and shown as functions of dimensionless interface height $z_0/H$ and dimensionless solitary-wave amplitude $A_w/(z_0-0.5H)$.

Figure 10

Predicted waveform from numerical solutions of the Gardner equation after a fixed amount of time as a function of the length along the tank $x$ and the scaled distance between the interface height and the middepth $(z_0-0.5H)/H$. For clarity of presentation, we have reversed the amplitude of the solutions so that the waves appear as waves of elevation.

Figure 11

Hovmöller plots of the horizontal velocity along the inviscid top boundary for the cases (a) Z32A-4, (b) Z32A0, (c) Z28A-4, and (d) Z28A0.

Figure 12

Horizontal velocity of (a) Z32A-4, (b) Z30A-4, (c) Z28A-4, and (d) Z26A-4 at $t=100$ s. The pycnoclines are indicated by the black contours and the waves' crests (determined by the locations of the maximum wave-induced horizontal velocity) are indicated by the dashed lines.

Figure 13

Same as Fig. 12 but for (a) Z32A0, (b) Z30A0, (c) Z28A0, and (d) Z26A0 at $t=100$ s.

Figure 14

Visualization of Table 5, with (a) amplitude, (b) half-width, (c) propagation speed, and (d) asymmetry of the leading waves versus the background shear at $t=100$ s. The asymmetry is measured using the ratio $(x_R-x_C)/(x_C-x_L)$, while the background shear is measured based on the seiche-induced vorticity ${\omega }_{max}$. Cases with similar background shear are indicated by the same symbols, with black (red) symbols indicating the existence of a solitary-like (dispersive) wave.

Figure 15

Stratification (measured by the pycnocline height $z_0$ scaled by the depth $H$) versus background shear (measured by ${\omega }_{max}$) at $t=100$ s, for simulations listed in Table 5. Black (red) symbols indicate cases in which a solitary-like (dispersive) wave exists. The dashed curve indicates the approximate boundary between the solitary-wave regime and the dispersive-wave regime.

Figure 16

Amplitude versus (a) half-width and (b) propagation speed for five different ISWs obtained by solving for the DJL equation in the background environment of the case Z26A-4 at $t=100$ s and $x=9$ m.