Chapter III: Mixing-driven mean flows and submesoscale eddies over mid-
3.5 Circulation and restratification in ridge flank canyons
Figure 3.7: Comparison of the cross-ridge velocity between the simulations with and without a canyon andin situobservation from the Brazil Basin Tracer Release Experiment. (a) Cross-ridge velocity in the three-dimensional simulation without canyon averaged over days 741–856, (b) cross-ridge (i.e. along-canyon) velocity in the three-dimensional simulation with canyon averaged over days 741–856, (c) com- parison of the along-canyon velocity profiles between the simulation and mooring observation. The black and magenta curves in (c) correspond to the sampling lo- cations marked in (b). The red dots denote the record mean along-canyon velocity from the mooring. The dotted and dashed lines roughly represent the canyon tops in the observations and the simulation.
submesoscale baroclinic eddies discussed in the previous section. In the simulation with a canyon, in contrast, a strong flow with a maximum velocity of over 1 cm runs up the ridge inside the canyon (Fig. 3.7b). This large up-ridge flow is banked slighted against the southern wall of the canyon and is accompanied by a return flow on the northern side of the canyon. We will explain the locations of the up- and down-ridge flows below. The up-ridge flows extend along the canyon axis across the entire ridge flank (Fig. 3.8a). Above the canyon, two larger-scale geostrophically balanced flows (∼ 0.005 m s−1) run in opposite directions. They are due to the prescribed mixing and the subsequent tilting of isopycnals on the canyon walls.
As part of the Brazil Basin Tracer Release Experiment in 1997, a mooring was
Figure 3.8: The cross-ridge flow and stratification evolution inside the canyon.
(a) The cross-ridge velocity averaged over days 694–856 along the center axis of the canyon (y = 50 km). (b) Time evolution of stratification in the center of the ridge flank and in the center of the canyon (x= 0 and y= 50 km).
deployed in a deep canyon on the western flank of the Mid-Atlantic Ridge near 22°S (Thurnherr et al., 2005; Toole, 2007). Eight current meters recorded velocities at half-hour intervals for a little over two years. We compare the record mean in situmeasurements with the simulated flow (again averaged over days 741–856) in the corresponding location in the middle of the ridge flank (x = 0) and inside our idealized canyon. Given the strong cross-canyon shear, we sample the simulated cross-ridge velocity with vertical profiles at two different locations: one in the center of the canyon (y =50 km) and one located in the return flow (y= 53 km, Fig. 3.7b).
The simulated up-ridge flow qualitatively agrees with the measured velocity profile, both in magnitude and in vertical structure (Fig. 3.7c). The simulated flow is weaker and more confined in the vertical than the observed flow, as might be expected because the canyon in the simulation is shallower than the about 1000 m deep canyon the mooring was placed in. The difference in the magnitude of the cross-slope flow is also likely due to the difference in the canyon width such that the observed canyon (∼10km wide) is narrower than the idealized canyon (∼20km wide), which has a difference in the blocking effects of the cross-canyon flows (this is explained later).
The presence of the strong return flow in the simulation warrants caution in inferring up-ridge transport from moored observations—the inferred transport in the canyon can depend sensitively on where exactly the mooring is placed.
The up-ridge flow inside the canyon is orders of magnitude larger than that in the simulations without a canyon (Section 3.4). As pointed out by Dell (2013), the canyon walls block along-ridge flow, preventing the across-ridge momentum that is generated by buoyancy forces from being deflected into the along-ridge direction. The cross-ridge flow in the canyon can be captured qualitatively by the one-dimensional system (3.7)–(3.10), modified by setting v = 0, or equivalently
f = 0.
We here extend Dell’s argument to the case with bottom-intensified mixing. We il- lustrate the impact of the blocking of the along-ridge flow by canyon walls by solving the one-dimensional system (3.7)–(3.10) for the steady-state flows and stratification with and without rotation. The equations are solved with the same procedures in Dedalus as in the temporal evolution case (section 3.4 and Fig. 3.4), with the time-dependent terms set to 0. The non-rotating solution represents a case in which the along-slope flow is blocked perfectly by the canyon walls; the rotating case is the same as that considered in Section 3.4a and corresponds to a planar slope. In the blocked case, an up-ridge flow spans the bottom 100 m and has a maximum in
Figure 3.9: Comparison between the one-dimensional solutions of the unblocked (rotating, f = −5×10−5s−1) and blocked (non-rotating, f = 0) cases. (a) Cross- ridge velocity, (b) along-ridge velocity, and (c) stratification. All curves show the steady-state solutions.
excess of 1 cm s−1, i.e. it is much larger than any cross-ridge flow in the rotating case (Fig. 3.9a). The bottom-intensification of the mixing drives a return flow above 100 m, which has a magnitude only somewhat weaker than the up-ridge flow. The up-ridge flow in the blocked case is captured to leading order by the approximate analytical solution (Callies, 2018)
u= −κ1cotθe−z/h
h +2qcotθ(κ0+κ1)e−qzsinqz, where q4= N2sin2θ 4(ν0+ν1)(κ0+κ1).
(3.14) The approximation leading to (3.14) is marginal because it requiresqh 1 while qh =4.4 for the parameters used here, but it still qualitatively captures the numerical solution.
The strong cross-ridge flow allowed by the along-ridge blocking is strong enough to maintain strong stratification (Fig. 3.9c). The stratification in the outer layer (above q−1 = 45 m) is close to the far-field value. The stratification is reduced only in the inner layer. This is in stark contrast to the rotating case, where stratification is reduced throughout the outer layer extending tohlogκ1/κ0 =700 m (Fig. 3.9c).
Here in this one-dimensional system, it is clear that the canyon wall blocks the development of the along-ridge flow, and thus enhances the cross-ridge flows inside
the canyons. This confirms the role of the canyon walls in developing the strong up-canyon flow in the three-dimensional simulation (Fig. 3.7). The return flow, as part of the dipole flow structure in response to bottom-enhanced mixing, has to sit on top of the strong up-slope flow since no other factors can break the symmetry in the cross-slope direction. We note that the shifted location of the return flow inside the canyon (Fig. 3.7b) is due to the turning of the cross-canyon flow in the local BBL system by Earth’s rotation, as explained later in this section.
Similar to the three-dimensional simulation without a canyon, the local stratification inside the canyon gets maintained (Fig. 3.8b). Thus, within the ridge-flank canyon, a strong stratification is maintained due to the mean up-ridge buoyancy advection, implying that the canyon walls have effectively turned the local BBL system into a non-rotating system, thus providing a different re-stratification mechanism compared with the flanks of the large-scale ridge.
Notice that there exists a layer of weak stratification out of the local BBL and higher up inside the canyon, which seems inconsistent with the measured stratifi- cation (Fig. 3.8b). After careful examination, we find that the maximum growth rate associated with the submesoscale baroclinic instability (Stone, 1966) inside this weakly stratified layer (withO(10)Richardson number) is 5.5×10−6s−1. The corresponding wavenumber is about 3.85×10−3m−1, dimensionalized using a char- acteristic velocity magnitudeU = 1 m s−1inside the canyon and f = 5.5×10−5s−1. The damping rate from the hyperviscosity used in the model at this wavenumber is 6.9×10−6s−1, of similar magnitude to the maximum growth rate. Thus, this layer of weak stratification is likely an artifact from the model such that the growth of submesoscale baroclinic eddies are damped by the hyperviscosity applied. We expect the layer to be more stratified once smaller scale baroclinic eddies are re- solved. On the other hand, this seemingly unrealistic weak stratification manifests the role of baroclinic restratification from submesoscale eddies inside the canyon in maintaining the local stratification.
As shown in the one-dimensional BBL system in Fig. 3.9a, the return (or downslope) flow sits on top of the much stronger up-slope flow, unlike the shifted return flow inside the canyon (Fig. 3.7b). A similar shift of the return flow was found in a previous simulation over a planar slope bounded by purely vertical sidewalls (Dell, 2013). This was explained such that weak cross-canyon flows turn to the left in the southern hemisphere and form the downslope flow on the northern canyon wall. To confirm that this explanation holds with the current sloping canyon wall
Figure 3.10: Flows in a two-dimensional canyon without the large-scale ridge.
(a) Cross-canyon flow at day 69, (b) along-canyon flow at day 69, (c) cross-canyon flow at day 579, (d) along-canyon flow at day 579. The shading shows velocities, black contours are isopycnals.
set-up, we designed a series of simulations to isolate different components in the three-dimensional simulation that might break this symmetry. We first ran a two- dimensional simulation with the large-scale ridge removed, in order to ensure that the return flow is part of the solution in the large-scale ridge system. The simulation results are shown in Fig. 3.10, where there is only symmetric along-slope flows in the canyon. In this local one-dimensional BBL system inside the canyon, the up-slope flows across the sloping canyon walls turn into the along-slope direction due to the rotation. Outside of the canyon, isopycnals tilt downward as a result of the background mixing and much larger geostrophically-balanced along-slope flows can be seen near the canyon mouth. Throughout the simulation, the symmetry in the along-slope flows across the canyon is maintained. This implies that the shifted return flow is part of the BBL system associated with the large-scale ridge slope.
However, there are still at least two candidates in the original three-dimensional
Figure 3.11: Simulation without the ridge flanks, which isolates the canyon effects.
(a) model topography, (b) cross-ridge flow at day 35. The shading shows velocities, black contours are isopycnals.
simulation that can break the symmetry and cause the shift of the return flow. The first one is the ridge flank. The along-ridge flow over the ridge flanks can flow into the canyon and interact with the up-canyon flows in the local BBL system above the canyon walls. Mass divergence/convergence can result in vertical velocity and tilt the isopycnals locally and thus break the symmetry across the canyon. The second candidate is Earth’s rotation that can deflect the up- and down-ridge flows. To isolate the impact from outside the canyon, we have extended the canyon walls all the way up to the top of the domain (Fig. 3.11a) such that there is no ridge flanks and thus no interaction between the flows from inside and outside the canyon. Upon checking on the distribution of the cross-ridge flow, we have found similar shift in the location of the return flow that is on the northern canyon wall (Fig. 3.11b). This confirms that the primary reason for the shift of the return flow is the Earth’s rotation rather than the presence of the ridge flanks and their associated mean flows.