Extreme bed shear stress during coastal downwelling

Abstract

The wind-driven circulation of coastal oceans has been studied for many decades. Using a 2.5-dimensional hydrodynamic model, this work unravels new aspects inherent with this circulation. In agreement with previous studies, downwelling-favorable coastal winds create an overturning cross-shelf circulation that operates to mix nearshore water. On timescales of days, this circulation tends to eliminate itself causing a “shutdown” of the cross-shelf circulation. For the first time, here, the author demonstrates that this shutdown is accompanied by creation of a zone of extremely high bed shear stresses (> 0.35 Pa) that operates to “plow” the seabed over an offshore distance of ~ 10–20 km. The author postulates that the associated sediment erosion episodes and their likely ammonification of the water column are key in the understanding of the biogeochemistry shaping coastal marine ecosystems.

Appendices

Simplified Turbulence Closure Scheme

1.1 Description

This additional study applies the same shelf model as in the main study, but with one modification. Instead of using the k-ε scheme, vertical eddy diffusivity, A_z, is diagnosed here from Kochergin’s turbulence closure (Kochergin 1987) that can be written as

$$ {A}_z={\left(c\Delta z\right)}^2\sqrt{{\left(\partial u/\partial z\right)}^2+{\left(\partial v/\partial z\right)}^2-{N}^2} $$

(A1)

where the free parameter is set to c = 0.2. Vertical eddy diffusivity, K_z, is again based on a turbulent Prandtl number of 0.7. The lower bound of A_z is set to a molecular value of 10⁻⁶ m²/s. The upper bound of A_z is set to 0.1 m²/s. A value of A_z = 0.05 m²/s is applied near the sea surface as a representation of background wind stirring of surface water.

Note that A_z in (A1) becomes negligibly small as the gradient Richardson number (19) approaches unity. Hence, additional treatment is required to parameterize dynamic instabilities such as convective or Kelvin-Helmholtz instabilities that are expected to occur when Ri_g < 1/4 (e.g., Baines and Mitsudera 1994; Cushman-Roisin and Beckers 2011). To account for this, the scheme is amended by the additional parameterization of effective eddy viscosity as a function of the local gradient Richardson number (e.g., Large et al. 1994; Wijesekera et al. 2003); that is;

$$ {A}_z=\left\{\begin{array}{c}{A}_{\ast }{\left[1-{\left({\mathrm{Ri}}_{\mathrm{g}}/{\mathrm{Ri}}_{cr}\right)}^2\right]}^3\ \mathrm{when}\kern0.5em 0<{\mathrm{Ri}}_{\mathrm{g}}<{\mathrm{Ri}}_{cr}\ \\ {}{A}_{\ast}\kern6.5em \mathrm{when}\kern0.5em {\mathrm{Ri}}_{\mathrm{g}}<0\end{array}\right.\kern0.5em $$

(A2)

Here, we use the theoretical value of Ri_cr = ¼ in conjunction with A_* = 0.1 m²/s in experiment E-1 and a reduced value of A_* = 0.01 m²/s in experiment E-2. Whenever Ri_g < Ri_cr the value from (A2) is always used to override that from (A1). Hereby it should be firstly noted that the choice of A_* = 0.1 m²/s follows from features of the mixing zone in the control experiment (see Fig. 6) and it is much larger than typically used to simulate turbulent diffusivity/viscosity in the interior of the water column (A_* ~ 0.005 m²/s). Secondly, it should also be explained that purpose of reducing A_* to 0.01 m²/s in the second experiment is to demonstrate that the high bed shear stresses developing near the downwelling front are the consequence of enhanced vertical momentum diffusion. Otherwise the experiments E-1 and E-2 are identical to the configuration of the control experiment (D-1, see Table 1).

1.2 Results

Experiment E-1 largely reproduces the results of the control experiment except for an overestimation of turbulent stirring near vertical boundaries (Appendix Fig. 14, compared with Fig. 6). This result is not unexpected given that, unlike in the k-ε scheme, the mixing length is not limited in vicinity of boundaries here. As a consequence both Ekman layers are thicker than in the control experiment. Nevertheless, all other scales are remarkable close to the control prediction, including the progression of bed shear stresses (Appendix Fig. 15, compare with Fig. 8). Again, the shelf model predicts the offshore progression of a peak bed shear stress slightly above 0.4 Pa that moves offshore at a rate of ~ 3 km per day. While experiment E-2 also predicts a similar progression due to the offshore displacement of the downwelling jet, the resultant maximum bed shear stresses are significantly smaller (Appendix Fig. 15), as least on time scales < 5 days. Hence, it is the shear instability process (induced by the cross-shelf circulation) that significantly enhanced bed shear stresses via vertical mixing of the along-shelf momentum. Quod erat demonstrandum.

High-Resolution Nonhydrostatic Simulations

1.1 Description

The shelf model used in this work cannot resolve the special scales of nonhydrostatic turbulent vortices inherent with the Kelvin Helmholtz instability mechanism which have an aspect ratio (ratio between horizontal and vertical scales) of unity. In order to resolve those scales, the hydrodynamic equations detailed in Section 2.1.3 are applied here with a finer grid spacing of Δx = Δz = 1 m on a smaller horizontal spatial scale.

The nonhydrostatic model is initialized by the predictions from the control experiment (D-1) using variable values for a selected single grid column (which has a horizontal width of 1 km) at a selected time. Hence, the nonhydrostatic model considers a 1-km wide model domain of a total depth corresponding to the location of the control experiment. This one-way coupling is done after every day of the “mother” simulation and at an interval of 2 km (i.e., for very second grid column of the shelf model) to an offshore distance of 30 km. Altogether this gives 10 × 15 = 150 simulations, but it is sufficient to only present the results of a few selected results. The total simulation time of nonhydrostatic model runs is 4 h using a numerical time step of Δt = 1 s. Initially small random fluctuations are added to the density field to seed minuscule fluctuations that can grow as part of instability processes. The vertical velocity field starts with zero values.

The nonhydrostatic assumes a flat seafloor (on the spatial scale of 1 km), and uses the same wind-stress forcing as the shelf model. Additionally the nonhydrostatic model also accounts for an external barotropic pressure-gradient force (due to the sloping surface), also prescribed from the shelf model. This pressure-gradient force is kept constant over the simulation period (5 h). Furthermore, the model uses cyclic horizontal boundaries, which ignores any lateral advection effects. The model also adopts Kochergin’s turbulence closure but without parameterization of shear-flow and convective instabilities, which are resolved in the model. For isotropic turbulence, Kochergin’s turbulence scheme can be expressed by (e.g., Kämpf and Backhaus 1998):

$$ {A}_x={A}_z={c}^2{\left(\Delta x\Delta z\right)}^2\sqrt{{\left(\partial u/\partial z\right)}^2+{\left(\partial v/\partial z\right)}^2+{\left(\partial v/\partial x\right)}^2+{\left(\partial w/\partial x\right)}^2-{N}^2} $$

(B1)

where c = 0.2. In addition, a turbulent Prandt number of unity is assumed (K_x = K_z = A_x = A_z). The aim of this supplementary study is to test whether the vertical shear of u in conjunction with the density stratification predicted by the shelf model can initiate Kelvin-Helmholtz instabilities and, if so, whether, this mechanism leads to the predicted enhancement of bed shear stresses via modulation of v. To illustrate any stirring mechanism, the nonhydrostatic model also predicts the evolution of a passive concentration field from the advection-diffusion equation:

$$ \frac{\partial }{\partial t}C+u\frac{\partial }{\partial x}C+w\frac{\partial }{\partial z}C=\frac{\partial }{\partial x}\left({K}_x\frac{\partial }{\partial x}C\right)+\frac{\partial }{\partial z}\left({K}_z\frac{\partial }{\partial z}C\right) $$

(B2)

which is of the same form as the density conservation Eq. (4). Initially, C varies linearly between zero and unity over the depth of the water column, using zero-flux vertical and cyclic lateral boundary conditions.

1.2 Results

Only results for the simulations that start from day 5 of the mother simulation are discussed here. Other start times yielded similar results. Within the stirring zone, for instance at x_o = 12 km (see Fig. 6), the nonhydrostatic model predicts the onset of shear-flow instabilities within 2–3 h of simulation (Appendix Fig. 16). The instabilities start to develop first near the seafloor before filling the entire water column. In contrast, outside the mixing zone at x_o = 30 km, the pronounced density stratification near the bottom of the water column (see Fig. 6) prevents turbulence generation in vicinity of the seafloor (Appendix Fig. 17). Hence, the bed shear stress remains at moderate levels.

Appendix Fig. 18 displays the evolution of bed shear stresses at x_o = 12 km. Initially of bed shear stress rapidly decreases uniformly in the entire mode domain over the first hour of simulation. This decrease is caused by a modified vertical eddy viscosity that leads to a decrease of the along-shelf velocity component v near the seafloor. After the onset of dynamic instabilities, which is apparent from the increase of the standard deviation of bed shear stresses, the maximum bed shear stresses “bounce back” to reach almost the same value (~ 0.4 Pa) as simulated by the shelf model.

Simulation for other offshore locations confirm that the entire zone of apparently high vertical eddy viscosity/diffusivity (see Fig. 6d) is prone to the onset of shear-flow instabilities and vigorous mixing in the entire water column (Appendix Fig. 18). The nonhydrostatic simulations also confirm that it is the downward mixing of long-shelf momentum v that substantially enhance bed shear stresses to extremely high values.

Fig. 14

Same as Fig. 6, but for experiment E-1

Fig. 15

Time series of bed shear stress (Pa) at selected offshore locations (x_o) for the experiments E-1 and E-2

Fig. 16

Nonhydrostatic model simulation. Spatial distributions of the concentration field C after a 1.6 h, b 2.5 h, and c 4.3 h of simulation. The model is initialized with values from the shelf model (control experiment) at x_o = 12 km after 5 days of simulation

Fig. 17

Same as Appendix Fig. 16, but initialized with values from the shelf model (control experiment) at x_o = 30 km. Shown are the distributions of C after a 1.6 h and b 6 h of simulation

Fig. 18

Results for the nonhydrostatic model simulation that is initialized with values from the shelf model (control experiment) after 5 days of simulation. a Time series of bed shear stress (Pa) at x_o = 12 km. The solid line shows the spatial average, the dashed lines account for twice the standard deviation (i.e., 95% confidence interval). Note that the upper curve roughly corresponds to maximum bed shear stresses. b Resultant average and range (based on twice the standard deviation) of bed shear stresses as a function of offshore distance x_o. Arrows indicate instances in which the nonhydrostatic model predicts the onset of shear flow instabilities. The red arrow (b) highlights the region of peak bed shear stresses