Numerical investigation of breaking internal solitary waves

High-resolution three-dimensional large-eddy simulations are used to investigate the effects of internal solitary waves (ISWs) breaking over a sloping boundary. The lock release method is applied in a two-layer stratified fluid system to generate three different breaking mechanisms (i.e., plunging, collapsing, and surging breakers). The different breaking types are investigated in terms of their effects on the dynamics of the ISW and the interaction of the ISW with the sloping boundary. During each breaking event, the pycnocline region entrains fresher water from the upper layer and saltier water from the lower one. The associated increase of the intermediate density layer also induces changes of the pycnocline water density. This process occurs with a velocity that can be evaluated using the bulk entrainment parameter. We show how the intermediate layer features depend on the ISW shoaling and breaking dynamics and we discuss entrainment in breaking ISWs. The instabilities induced by boundary layer separation allow entrainment of saltier water, while the run-up of the gravity current causes the decrease of the intermediate layer mean density. Simultaneously, the entrained water mixes into the pycnocline region. For all cases, the temporal evolution of the instantaneous mixing efficiency is discussed. The plunging breaker case shows the largest amount of mixing, which is mostly induced by rear-edge overturning in the onshore direction. The largest entrainment is observed in the surging breaker case in response to the large gravity current flowing upslope. The paper discusses how the different turbulent instabilities induced by the ISWs breaking affect the time delay between the times when entrainment of patches of salty and fresh water from the neighboring layers occurs and the time the density of the intermediate layers becomes fairly uniform via mixing. We finally point out that the entrainment parameter and the mixing efficiency describe two different effects of the turbulent instabilities occurring in a stratified fluid in terms of changes of the bulk density profile.

Figure 1

Sketch of the domain used to study breaking ISWs on slopes. The dashed line represents the initial pycnocline position in the two layer-stratified fluid. The black dashed vertical line shows the initial position of the lock gate.

Figure 2

Snapshots in time visualizing the heavier fluid in the experiments (on the left) and the nondimensional density fields obtained by numerical simulation (on the right) for the collapsing case (case 2) (a) after the experimental or numerical initial setting ($t^{\prime }/T=-5.1$), (b) a short time after the gate is removed ($t^{\prime }/T=-4.51$), (c) after the ISW formation ($t^{\prime }/T=-3.43$), (d) during the ISW propagation toward the sloping boundary ($t^{\prime }/T=-2.34$), and (e) before the interaction of the ISW with the sloping boundary ($t^{\prime }/T=-1.23$). Yellow and red lines visualize the $2\%$ and $98\%$ isopycnal surfaces, respectively.

Figure 3

Snapshots in time visualizing the heavier fluid in the experiments (on the left) and the nondimensional density fields obtained by numerical simulation (on the right) for the collapsing case (case 2) (a) at the time at which the ISW starts interacting with the sloping boundary and the horizontal velocity in the upper layer starts to decrease because of the presence of the sloping boundary ($t^{\prime }/T=0$); (b) during the bolus generation, point $A$ in the following figures ($t^{\prime }/T=1.7$); (c) at a time when mixing induced by the bolus is significant, point $B$ in the following figures ($t^{\prime }/T=2.05$); (d) after the gravity current running up the slope forms, point $C$ in the following figures ($t^{\prime }/T=2.96$); and (e) at the time the gravity current reaches the end of the slope, point $D$ in the following figures ($t^{\prime }/T=5.59$). Yellow and red lines characterized the $2\%$ and $98\%$ isopycnal surfaces, respectively.

Figure 4

Case 2 experiment and numerical simulation snapshot with the field of view showing the entire domain. (a) Experimental normalized grayscale field $g_n$ and (b) numerically predicted dimensionless density field $\langle {\rho }^{*}\rangle$ in a horizontal plane ($x/H\text{−}{t}^{\prime }/T$) situated at fixed depth ($y/H=0.74$) from the bottom. (c) Normalized grayscale field and (d) numerically predicted dimensionless density field in the vertical plane ($t^{\prime }/T\text{−}y/H$) at the middle of the domain ($x/H=5.35$).

Figure 5

Nondimensional density fields predicted by numerical simulation for (a)–(e) the plunging case (case 1) and (f)–(j) the surging case (case 3). Shown for the plunging breaker are (a) the wave approaching the slope ($t^{\prime }/T=0$); (b) steepening of the trailing edge ($t^{\prime }/T=1.03$); (c) mixing induced by the trailing edge overturning, point $E$ in the following figures ($t^{\prime }/T=2.46$); (d) gravity current flowing upslope, point $F$ in the following figures ($t^{\prime }/T=7.44$); and (e) the gravity current stopping ($t^{\prime }/T=13.48$). Shown for the surging breaker are (f) the wave approaching the slope ($t^{\prime }/T=0$); (g) small instability for boundary layer separation, point $G$ in the following figures ($t^{\prime }/T=1.17$); (h) formation of the gravity current ($t^{\prime }/T=1.84$); (i) the gravity current flowing upslope, point $H$ in the following figures ($t^{\prime }/T=2.25$); and (j) the gravity current stopping ($t^{\prime }/T=3.42$).

Figure 6

Analysis of the entrainment process. (a) Case 2, nondimensional density. The arrows identify the direction of the entrainment which occurs perpendicularly to the $2\%$ and $98\%$ isopycnal surfaces (yellow and red lines, respectively). Also shown are the temporal evolution of (b) the dimensionless volume and (c) the averaged density of the pycnocline water. Results are shown for case 1, the plunging breaker (solid lines); case 2, the collapsing breaker (dashed lines); and case 3, the surging breaker (dash-dotted lines). Points $A$–$H$ are defined in Figs. 3 and 5.

Figure 7

Temporal variation of bulk entrainment parameter for (a) the collapsing case, (b) the plunging case, and (c) the surging case. The dashed lines show the EP obtained by using the $2\%$ isopycnal surface (entrainment of fresh water), the green solid lines show the EP obtained by using the $98\%$ isopycnal surface (entrainment of salty water), and the black solid lines show the EP obtained by using both the $2\%$ and $98\%$ isopycnal surfaces (entrainment of the fresh water and salty water into the intermediate density layer). Points $A$–$H$ are defined in Figs. 3 and 5.

Figure 8

(a) Temporal variation of volume-integrated energy for the collapsing case. The kinetic energy $E_k$, total potential energy $E_p$, background potential energy $E_b$, and available potential energy $E_a$ are all normalized by $|E_{a,\text{min}}|$. (b) Normalized rate of change of background potential energy for the collapsing case. Points $A$–$D$ are defined in Fig. 3.

Figure 9

Instantaneous mixing efficiency $\eta$ and bulk mixing efficiency ${\eta }_m$ (dashed lines) for (a) the collapsing case, (b) the plunging case, and (c) the surging case. Points $A$–$H$ are defined in Figs. 3 and 5.