Internal solitary wave bottom boundary layer dissipation

Boundary layer instability beneath internal solitary waves (ISWs) of depression may be a significant source of wave-energy dissipation and drive localized mixing and resuspension in coastal regions. Wave flume experiments were undertaken to measure the dissipation of turbulent kinetic energy in the boundary layer beneath ISWs shoaling over a flat bottom. The rate of dissipation of turbulent kinetic energy $(\varepsilon \sim {10}^{-7}-{10}^{-2}\mathrm{W}\mathrm{k}{\mathrm{g}}^{-1})$ was elevated within the unstable boundary layer region, occurring for waves with momentum thickness-based Reynolds number ${\mathrm{Re}}_{ISW}>\sim 200$. Dissipation was parametrized in terms of wave energy $(\mathrm{RMSE}=10\times {10}^{-4}\mathrm{W}{\mathrm{m}}^{-1})$ and ${\mathrm{Re}}_{ISW}$ $(\mathrm{RMSE}=7\times {10}^{-4}\mathrm{W}{\mathrm{m}}^{-1})$. The dissipation length scale was computed from wave-integrated boundary layer dissipation $L_d=\frac{cE}{D}$ and parametrized as a function of wavelength $\lambda$ and ${\mathrm{Re}}_{ISW}$ as ${L_d}^{*}=100\lambda +2.5\times {10}^{10}\lambda {\mathrm{Re}}_{ISW}^{-3.7}$ $(\mathrm{RMSE}=\sim 50\lambda )$. Unstable waves had a dissipative length scale of $\sim 100\lambda$, in agreement with limited field observations, whereas stable waves propagated significantly further $(>\sim 1000\lambda )$.

Figure 1

Schematic of the experimental setup, showing both the initial condition behind the gate and the resultant wave. ISWs of depression were generated within the two-layer fluid by suddenly lifting the gate [24].

Figure 2

(a) Example of logarithmic fit (line) to ADV-observed instantaneous velocity (circles) for experiment 1. (b) The ratio between kinetic energy, $KE$ and available potential energy, $APE$ computed from the DJL equation ($KE\sim 1.15APE$) vs wave Reynolds number, $\mathrm{Re}{}_W=a{c}_o/\nu$. The measured instantaneous near-bed velocity profiles, required to compute $\varepsilon$, show a nonphysical comparison to law-of-the wall solutions, within 5 mm above the bed, where the signal-to-noise ratio was poor. M07 and K06 refer to Moum et al. [51] and Klymak et al. [43].

Figure 3

Time series of a stable wave (experiment 9). (a) Instantaneous streamwise velocity, $u$; (b) mean horizontal velocity, $U$; (c) mean vertical velocity, $W$; and (d) rate of dissipation, $\varepsilon$ (${\log }_{10}$ scale). The near-bed jet flow is shown in blue in panels (a) and (b).

Figure 4

Time series of an unstable wave (experiment 13). (a) Instantaneous streamwise velocity, $u$; (b) instantaneous spanwise velocity, $v$; (c) instantaneous vertical velocity, $w$; and (d) rate of dissipation $\varepsilon$ (${\log }_{10}$ scale). The near-bed jet flow has become unstable beneath the rear shoulder of the wave [blue in panel (a)].

Figure 5

(a) Stable and unstable waves in $a/H$ vs $\mathrm{Re}{}_W$ space showing laboratory observations (present study and Carr et al. [36]; C08). The solid line is the fit from the Diamessis and Redekopp [54] (D06) 2D simulations. (b) Stable and unstable waves in $\mathrm{Re}{}_{ISW}$ vs $P_{ISW}$ space showing laboratory observations, numerical simulations, and field observations (Table 2). The solid line is the fit from Aghsaee et al. [38] (A12) 2D numerical simulations. (c) Detail of laboratory data in panel (a). (d) Detail of laboratory data in panel (b).

Figure 6

(a) Decrease of wave energy, $E/E_i$ against the nondimensional distance traveled for stable [experiment 1; $E/E_i=-0.001\left(\frac{x}{H}\right){\left(\frac{h_2}{H}\right)}^2+1$] and unstable [experiment 16; $E/E_i=-0.002\left(\frac{x}{H}\right){\left(\frac{h_2}{H}\right)}^2+1$] ISWs. Also shown are field data from Moum et al. [51] (M07) and Michallet and Ivey [21] (M&I99). $E$ is the total energy scaled by the initial total energy $E_i$ at location $x$. (b) Observed dissipation $D$ [Eq. (8)] compared to total wave energy $E$. (c) Observed dissipation $D$ compared to momentum thickness Reynolds number $\mathrm{Re}{}_{ISW}$. (d) Decay length scale [Eq. (9)] normalized by wavelength and compared to momentum thickness Reynolds number $\mathrm{Re}{}_{ISW}$. Shroyer et al. [37] proposed $L_d\sim 100\lambda$ [Eq. (10)] and the best fit is given by Eq. (11).

Figure 7

Comparison of integrated dissipation [$D$ from Eq. (8)] to dissipation parametrizations $D^{*}$ based on $E$, $\mathrm{Re}{}_{ISW}$, and $L_d^{*}$. A dotted 1:1 line is shown. Stable and unstable waves are shown with hollow and filled symbols, respectively. Magenta circles, red triangles, blue diamonds, and green squares correspond to the same symbols and colors as shown in Figs. 6 and 8.

Figure 8

Comparison of parametrized dissipation length scale $L_d^{*}$ [Eqs. (10, 11, 12, 13)] to dissipation length scale $L_d$ from Eq. (9) using the measured dissipation $D$. Data are normalized by the wavelength $\lambda$. A dotted 1:1 line is shown. Stable and unstable waves are shown with hollow and filled symbols, respectively. Magenta circles, red triangles, blue diamonds, and green squares correspond to the same symbols and colors as shown in Figs. 6 and 7.