Gravity currents down canyons: effects of rotation

Abstract

The flow of dense water in a V-shaped laboratory-scale canyon is investigated by using a non-hydrostatic numerical ocean model with focus on the effects of rotation. By using a high-resolution model, a more detailed analysis of plumes investigated in the laboratory (Deep-Sea Res I 55:1021–1034 2008) for laminar flow is facilitated. The inflow rates are also increased to investigate plume structure for higher Reynolds numbers. With rotation, the plumes will lean to the side of the canyon, and there will be cross-canyon geostrophic currents and Ekman transports. In the present study, it is found that the cross-canyon velocities are approximately 5 % of the down-canyon velocities over the main body of the plume for the rotational case. With rotation, the flow of dense water through the body of the plume and into the plume head is reduced. The plume head becomes less developed, and the speed of advance of the head is reduced. Fluid parcels near the top of the plume will to a larger extent be left behind the faster flowing dense core of the plume in a rotating system. Near the top of the plume, the cross-canyon velocities change direction. Inside the plume, the cross-flow is up the side of the canyon, and above the interface to the ambient there is a compensating cross-flow down the side of the canyon. This means that parcels of fluid around the interface become separated. Parcels of fluid around the interface with small down-canyon velocity components and relative large cross-canyon components will follow a long helix-like path down the canyon. It is found that the entrainment coefficients often are larger in the rotational experiments than in corresponding experiments without rotation. The effects of rotation and higher inflow rates on the areal patterns of entrainment velocities are demonstrated. In particular, there are bands of higher entrainment velocities along the lateral edges of the plumes in the rotational cases.

Appendices

Appendix A—Comparisons with Fig. 9a and Fig 12 in Darelius (2008)

In the Appendix, the density interface positions given as Fig. 9 and the velocity profiles given as Fig. 12 in Darelius (2008) are used in model-data comparisons. The numerical results are produced with Δx = 5.0 mm, Δx = 2.5 mm, and Δx = 1.25 mm and for P r = 7 and P r = 100, see the discussion in Section 2.1.

The density interface positions from the numerical results are compared to the interface positions from the laboratory experiments of Darelius (2008) for slope equal to 0.04 in Fig. 16. The values of ρ ^′ at the interface in Fig. 9 of Darelius (2008) are not specified, which makes a clean one to one comparison difficult. However, in Fig. 16, the interface positions for the model results after 300 s are given for ρ ^′ =‘0.45 kg m⁻³. The numerical results reproduce basic features of the flow presented in Darelius (2008), but there are quantitative differences that may be due to differences in the set up of the laboratory and numerical experiments, numerical errors, and/or errors in the data processing of the digital photos. It may be noted that the modeled density interfaces become more aligned with the laboratory interface for P r = 100. Furthermore, it may be seen that the interface positions for the numerical experiments with Δx = 1.25 mm and Δx = 2.5 mm are almost identical.

Fig. 16

The density interface position for slope equal to 0.04 from Fig. 9a of Darelius (2008) (solid line) and the interface positions of the $\rho ^{\prime }$ = 0.45 kg m⁻³ surface after 300 s from the numerical experiments (dashed lines). The results for P r = 7 are given to the left and the results for P r = 100 are given to the right

In order to compare velocity profiles from the numerical experiments to the measured profiles given in Fig. 12 in Darelius (2008), this figure is given here as Fig. 17. These profiles are for Q = 3.0 × 10⁻⁶ m³ s⁻¹ and ρ ^′ = 2.0 kg m⁻³ (twice the values in RUN2). The RUN2 experiments are repeated with these values of Q and ρ ^′, and time mean values of the along- and cross-canyon velocity components for the three horizontal grid sizes are given in Fig. 18. The corresponding profiles of time mean density are given in Fig. 19. The time means are taken over the period from 200 to 300 s. By comparing the velocity profiles in Figs. 17 and 18, it is seen that the order of magnitude of the maximum flow speeds are similar. The flow is, however, more uniform over the body of the plume in the laboratory experiment. In the numerical results, there are clear velocity maxima above the bottom boundary. The vertical profiles of numerical results show there are differences in the profiles produced with Δx = 1.25 mm and Δx = 2.5 mm, and it can therefore not be concluded that converged solutions are obtained with Δx = 1.25 mm. The differences between these two sets of solutions may indicate the order of magnitude of the errors in the results produced with the finest grid. From Figs. 18 and 19, it may be noticed that the numerical velocity and density profiles over the bodies of the plume are more consistent on the side of the plume than in the center of the canyon, even if the bottom is smoothed to avoid artifacts. When using P r = 100, the dissipation is reduced and this lowers the center of mass, see Fig. 19 and affect the vertical velocity profiles accordingly, see Fig. 18.

Fig. 17

Velocity profile a the center of the canyon and b 1 cm to the left (looking up the canyon) as measured with laser Doppler velocimetry (LDV) in experiment (i) of Table 2 in Darelius (2008). The theoretical velocity profiles from Ekman theory are shown in black. Courtesy of Elin Darelius

Fig. 18

Time mean velocity profiles from the numerical experiments corresponding to the LDV (i) experiments in Darelius (2008). Values from the center of the canyon are given to the left and from 1 cm to the left of the center of the canyon are given to the right. Values for P r = 7 are given in the top panel, and values for P r = 100 are given in the bottom panel. The front passes after approximately 80 s and the time means are taken over the period from 200 to 300 s

Fig. 19

Time mean density profiles from the numerical experiments corresponding to the LDV (i) experiments in Darelius (2008). Values from the center of the canyon are given to the left and from 1 cm to the left of the center of the canyon to the right. Values for P r = 7 are given in the top panel, and values for P r = 100 are given in the bottom panel

Appendix B—The Ekman spiral

In this appendix, the ability of the bottom boundary conditions given by Eqs. 2 and 3 to produce an Ekman spiral consistent with theory near the bottom is investigated. We consider the case discussed in Chapter 5.2 of Cushman-Roisin (1994). The time-dependent equations are

$$ \frac{\partial u}{\partial t} - f v \, = \, \nu \frac{\partial^{2} u}{\partial^{2} z}\, , \, \text{and} $$

(9)

$$ \frac{\partial v}{\partial t} + f (u- \overline{u}) \, = \, \nu \frac{\partial^{2} v}{\partial^{2} z}\, . $$

(10)

A uniform flow $\overline {u}$ is assumed in the ambient and $\overline {u}$ is set to 0.01 ms⁻¹. The constant viscosity ν is set to 1.0 ×10⁻⁶ m ² s⁻¹. The Coriolis parameter f is set to 0.76 s⁻¹. The vertical range is z = [0,H], and H is set to 0.25 m (the depth of the tank in the experiments). The initial values for u and v are u(z, t=0) = $\overline {u}$ and v(z, t=0) = 0 for z ∈[0,H]. At the surface, z = H, a Neumann boundary condition is assumed. That is, $\frac {\partial u}{\partial z}$ = $\frac {\partial v}{\partial z}$ = 0. The stresses at the bottom are given by Eqs. 2 and 3. The period of free oscillations is T = 2π/f. Numerical approximations after 10T are compared to the exact solutions given in Cushman-Roisin (1994) in Fig. 20. The exact solutions are for the steady state case and for an infinitely deep ocean. It is still found that the two sets of solutions agree very well.

Fig. 20

Vertical profiles of numerical approximations to u and v (U _M and V _M ) and exact vertical profiles U _E and V _E from Cushman-Roisin (1994)