Island Wakes in Shallow Water

Island Wakes in Shallow Water Changming Dong, James C. McWilliams, et al Institute of Geophysics and Planetary Physics, University of California, Los Angeles 1

ABSTRACT As a follow-up work of Dong et al (2007) on deep-water island wakes, we continue to investigate island wakes in shallow water using the Regional Oceanic Model System. The deep water implies the bottom stress can be neglected while the inhomogeneity in the bottom stress in shallow water should play an important role in the vorticity generation in island wakes. A series of numerical experiments are performed to study the wake formation and evolution. It is found that the vertical structure in the shallow water wake is significantly different from that from the deep-water wake due to the presence of the density frontal jet which results from the interaction between the stratification and bottom topography. The frontal jet reaches its maximum within the bottom boundary layer over the shelf break, which gives rise to the vorticity in addition to that from the lateral stress. The PV balance analysis exposes the frictional and diapycnal processes plays different roles in the PV anomalies. With the absence of the lateral stress, i.e., a sea mountain case, the surface vorticity becomes much weaker than that with the island. 2

1. Introduction With the presence of islands in the ocean where the rotation and stratification can not be neglected, the physical processes in lee of islands could be significantly influenced, which could be categorized into two types: the one is the oceanic response to the wind wakes, and the other is the oceanic current passing an obstacle, or referred to oceanic current wakes afterwards. The realistic island wakes could be very complicated due to the nonlinear combination of the above two types. For the former one, when a wind passes an island, the intensity of the wind behind of the island could decrease dramatically due to the increase in the land surface roughness and the blocking by high mountains over the island, which leads to the positive and negative wind curls formed on the right and left sides when one faces downstream of the wind, respectively. Ekman pumping could leads local upwelling or downwelling and even the formation of cyclonic or anticyclonic eddies. Basterretxea et al (2002) shows a robust observational evidence for the argument. For the latter one, oceanic current wakes can be categorized into two types considering different vorticity generation mechanisms (Tomczak, 1988): deep-water and shallowwater island wakes. There are three possible vorticity sources: 1) the lateral stress; 2) the bottom stress and 3) the tilting of the baroclinic flow. If the primary vorticity comes from the lateral stress, the island wake is considered as a deep-water island wake; when from 3

the bottom stress, it belongs to the shallow-water wake, where the horizontal vorticity could be tilted into the vertical component by baroclinic processes (Smolarkiewicz and Rotunno, 1989). Dong et al (2007) presented a series of numerical experiments to discussion the deepwater island wakes, where the vorticity generation from the lateral stress, its evolution and its sensitivity to a number of non-dimensional parameters (Reynolds number (Re), Rossby number (Ro) and Burger number (Bu)) are discussed in detail (for the definition of the non-dimensional parameters, please refer to Dong et al (2007). It is found that the vorticity generation in the deep-water island wakes in a stratified and a rotating fluid has a pattern of its sensitivity to the Re number similar to the classic fluid with homogenous and non-rotating fluid. The Strouhal number (St), which represents the eddy shedding frequency, is surprisingly similar to that in the classic fluid, i.e. 0.2. However, with the rotation, the occurrence of centrifugal instability (or inertial instability) could lead to the asymmetry in the anticyclonic and cyclonic eddies in the wake. Different ranges of the Re number forces the wake into different dynamic regime with varying scale of the Ro/Bu, i.e., different asymmetry in the cyclonic and anticyclonic eddies: when the small (large) Re number, anticyclonic (cyclonic) eddies dominates over cyclonic (anticyclonic) eddies with higher Ro/Bu. This conclusion agrees with laboratory experiments by Perrets et al (2006). Besides the centrifugal instability, the baroclinic and barotropic instability also take place in the deep-water wake. 4

In this paper, as a sequent work to Dong et al (2007), we discuss the shallow-water wakes using numerical experiments, where how the vorticity is generated and the vorticity evolve are explored. The difference among the deep-water wake and sea mountain (without the lateral stress) is discussed. The rest of the paper is organized in the following: Section 2 is the numerical configuration; A baseline experiment is discussed in Section 3. The parameter sensitivity is investigated in Section 4. Section 5 is the discussion, where the differences between the shallow-water, and deep-water and a flow passing a seamountain (with no lateral stress) are presented. Section 6 is a summary. 2. Model Configuration Dong et al (2007) applied the Regional Oceanic Model System (ROMS) to study the deep-water island wake where the bottom stress is neglectable. In this study, the ROMS is employed to the shallow-water island wake, where the bottom stress plays an important role in the vorticity generation and evolution. The same vertical profiles of the density and the incoming current as the baseline experiment in Dong et al (2007) are used in the present numerical experiments. The water depth is not uniform for but with a shelf slope, which is described in the formula: = 0 m, r r 0 h (x, y) = { (1) = (h max + h min )/2 + (h max -h min )/2 * tanh ( (r r break )/r width ), r > r 0 5

where (x0,y0) is the center of the island, r = [(x-x 0 )2+(y-y 0 ) 2 ] 1/2 is the distant from the center of the island, h max and h min are the deepest water depth in the domain, r break is the distance of the shelfbreak from the center of the island, and r width is the shelf width. The turbulence model KPP is used in the experiment, see Blaas et al (2006). As stated in Dong et al (2007), the background horizontal viscosity could be set zero because the implicit viscosity exists due to a biased upstream advection scheme used in the ROMS (Shchepetkin and McWilliams, 1998), for a detailed discussion of the implicit viscosity, please refer to Dong et al (2007). The solid boundary around the island has a zero-normal and no-slip flow implemented through a standard land-mask algorithm (Shchepetkin and O Brien, 1995). The same open boundary condition as Dong et al (2007) is applied: at the northern and southern sides, slippery tangential and zero-normal boundary conditions are applied while a clamped condition for the outgoing current and density profile with a sponge layer at the eastern downstream outflow boundary. The initial boundary condition for the entire domain is set equal to the upstream boundary condition except at the island points with island points with land masks. 3. Baseline Experiment In the baseline experiment, as indicated in Sect 2, the same surface-intensified vertical profiles of density and incoming current as Dong et al (2007) are used, see Fig. 1. The bottom topography is set as the island radius r 0 = 5km the shallowest water depth h min = 50 m, the deepest water depth offshore h max = 500m, the shelf break width is r break = 6

0 50 100 150 200 Depth (m) 250 300 350 400 450 500 0 20 40 60 80 100 120 Cross shelf Distance (km) Fig. 1 Vertical profiles of incoming flow (upper left) and density (upper right). The bottom panel is the topography of an island: the solid line is one with a slope for the baseline experiment, the thin dashed line is the one used for the deep-water wake by Dong et al (2007) and the thick dashed line is the one which has the radius as the width of the shelfbreak. 7

20km, and the shelf width is r width = 8km, see Fig. 1. The quadratic bottom stress is applied, τ = C d U b U b, where the C d = 2.5 x 10-3. The baseline case is referred as Case 1, see Table 1. To test the sensitivity of the numerical solution to the setting of the configuration, a series of numerical experiments are conducted, See Sect 4. Table 1 Numerical Experiments Bottom stress Island Radius With/Without Shallowest C d Shelf Slope Water Depth Case 1 2.5 x 10-3 5 km Yes 50m Case 2 0 5 km Yes 50m Case 3 2.5 x 10-3 5 km Yes 25m Case 4 2.5 x 10-3 5 km Yes 75m Case 5 2.5 x 10-3 5km Yes 100m Case 6 2.5 x 10-3 5km No 500m Case 7 2.5 x 10-3 10 km No 500m Case 8 2.5 x 10-3 20 km No 500m 3.1 Vertical Structure At Day 20, the relative vorticities at four levels are plotted in Fig. 2: surface (5m), the top of the shelf (50m) and the shelf break level (100m) and the middle-slope level (150m). It clearly shows that the relative vorticity does not reach maximum at the surface but at 8

shelf break levels though the incoming flow is surface intensified, see Fig. 1. At the surface (Level 5m), the positive vorticity is generated from two locations: the one is near island and the other is a few kilometers away from the island on the southern side of the island, while the negative vorticity comes from the layer near the island on the northern side. On Level 50m, the vorticities on both sides of the island are mainly generated from a layer a few kilometers away from the island. Within the upper layer above the shelf break (5m and 50m), the intensity of vorticity is relatively weaker than the lower layers (100m and 150m) which are within the shelf slope. Another feature is: on the southern side of the island, a positive vorticity sheet other than eddies is formed on the southern Fig. 2 The snapshot of the relative vorticity at four levels from the baseline experiment (Case 1) on Day 20. side of the island, but anticyclonic eddies develop on the northern side. The intensity of 9

the positive vorticity sheet is stronger than that of the anticyclonic eddies. Within the shelf slope levels, the vorticity comes from the lateral boundary layer and their intensity become much stronger than that within the upper layer above the shelf break. The vertical structure of the island wake with the shelf slope is different from that in the deep-water island wake, see Fig. 4 in Dong et al (2007), where the relative vorticity monogically (?) decreases with depth, in other words, the wake has the similar pattern as the incoming flow, i.e., the surface intensified. Figure. 3. The snapshot of the vertical profiles of the density and along-island (eastward) current at the section cross the island center. 10

To interpret the difference between the shallow-water and deep-water wakes, we examine the hydrographic data around the island. Fig. 3. When a stratified current passes an island with a shelf slope, the mixing due to the bottom stress tries to homogenize the density, which results in the horizontal density gradient within the bottom mixing layer. Since the bottom mixing is proportional to the bottom depth, the sharp change in the water depth gives rise to the formation of the front near the shelf break. An along-frontal jet is accompanied to the front due to the geostrophic balance. It is noted that the jet is a subsurface jet, and its generation is not from the lateral stress but from the non-uniform bottom mixing, which only takes place around the island. This is a baroclinic process. On the other hand, due to the geostrophic constrain, the flow follows the bottom topography and it will be sharpened at the shelf break, which gives the second mechanism for the formation of the along shelfbreak jet, which is a barotropic process. That is why we see the large vorticity ( the northern-gradient of eastward current) at a few kilometers away from the island from the surface to the bottom. Though the flow is restrained from reaching the island, the horizontal shear still develops around the island above the shelf, which generates the vorticity within the lateral boundary layer near the island. When the jet accompanied the density front induces the downwelling across-section current, it will intensify the front and the jet itself, that is what we see on the northern side of the island. When the jet introduces the upwelling cross-section circulation, it changes the density gradient by moving the dense water below and the jet becomes weaker and even reverse its direction (Chapman and Lentz, 2005), that is what happens 11

on the southern side of the island. The process gives rise to the asymmetry between the southern and northern sides of the island. Through the above analysis, one can see clearly the interaction between the bottom topography, stratification, bottom stress and lateral stress. The relative vorticity, which only involves the velocity field, can not fully represent the complicated processes involved in the wake formation. The Ertel potential vorticity is applied in next section. 3.2 Potential Vorticity Balance The Ertel potential vorticity (PV) is defined as (2) The PV equation is. where the frictional force F = (X, Y), (3) (4) 12

showing how the frictional torques on isopycnal surfaces or gradients of diapycnal mixing in the direction of absolute vorticity (PV) change (Marshall and Nursers, 1992). (5) Fig. 4. Distribution of the Potential Vorticity (PV) anomalies at four levels on Day 20 from the baseline case (Case 1). Fig. 4 plots the horizontal distribution of PV anomalies on Day 20. The anomalies are calculated relative to the upstream PV before it is disturbed by the island. It is seen that 13

the PV increase above the shelf break level (Level 10m and 50m) but decreases within the shelp slope levels and increase again below the below Level (200m). The PV Eq. (3) can be rewritten as where ADV is the advection term, FRIC is the frictional term, DIA is the diabatic term, and PRESS is the pressure gradient term, which is the numerical error (Thomas, 2006). (6) To calculate the PV balance, the methodology developed by L. Thomas (2007) is used to estimate each term in Eq. (6). A water volume which is bounded by the isotherm 18.5 and 22.5, the edge island, 50km upstream and downstream, and 30km south (north) of the center of the island is chosen for estimate the PV balance in the southern (northern) part of the island. Fig. 5 the isotherm lines (18.5 and 22.5, the thick black lines) are chosen for the volume to estimate the PV balance. 14

Fig. 6. The PV balance with the time: the southern volume (upper) and the northern volume (lower). In the northern volume, the diapycnal process is dominant over the frictional proceses and in the southern volume, the frictional term dominant. 15

4. Sensitivity In this section, the sensitivities of the numerical solution to the bottom stress, water depth and the configuration are discussed. 4.1 Bottom Stress As we have seen in Sect 3, the bottom stress plays an important role in the vorticity generation. In Case 2, the bottom stress is set to be zero, i.e., slippery bottom. We discuss the effect of the bottom stress on the PV. According to the Ertel PV definition, it is composed of three terms, which are related to the vertical density gradient, y-direction density gradient and x-direction density gradient: Each term at the vertical section cross the island center is plotted in Fig. 7. The vertical term is the dominant term in the PV, and the downstream gradient term (x-direction) is neglected. It can seen that without bottom stress, the PV anomalies are barely seen on the southern side of the island, while the large negative anomalies around the island are presented in Case 1 where the bottom stress is applied. For the free slippery bottom condition, the vertical and y-direction density gradient terms are compensated somehow. With the bottom stress, the y-direction density gradient term causes the negative PV anomalies on both northern and southern sides of the islands. 16

Fig. 7 Comparison of each term in PV between the slippery bottom (Case 2, upper panel) and frictional bottom (Case 1, lower panel). The first, second, and third columns are the term related to the vertical, y-direction, x-direction density gradients, respectively (from the left to the right). The most right column the total PV. 4.2 Water Depth Since the density front in the bottom boundary layer is resulted from the interaction of the stratification and bottom stress, the frontal formation is the change in the water depth. When the shallowest water depth is 25m, the main vorticity is generated not from the lateral stress but from a layer a few of kilometers away from the island, which is, as we see from the preceded section, the frontal jet over the shelf break. The intensity of 17

vorticity generated from the off-touched layer decrease with the water depth of the shelf increasing. When h min = 100m, the primary vorticity is generated from the lateral boundary at the surface, which converges to the deep-water island wake. Fig. 8 The snapshots of surface relative vorticity distribution with different shallowest water depth (h min ) on Day 20: h min =25m (Case 3), 50m (Case 1), 75m (Case 4) and 100m (Case 5). 4.3 Sea Mountain 18

As we have seen, the vorticity is generated by two processes: the one is the lateral stress and the other is the bottom stress. In the deep-water island wakes (Dong, et al 2007), we have seen the solution with the absence of the bottom stress. In the preceded sections, we discussed the cases with both mechanism presented. Here we show the case with the absence of the lateral stress: the island is sunken into the water, i.e., a sea mountain case (Case 6). With the absence of the lateral stress, it is obvious that the vorticity is much Fig. 9 The comparison of the shallow-water wake (upper) with the case with a sea mountain (lower, Case 6) weaker than that with the lateral stress. The main vorticity is generated from the frontal jet where the current shear is presented. Due to the frontal jet on the northern side is 19

constrained within the bottom boundary layer, the surface negative vorticity is much weaker than the positive vorticity. 5. Summary A series of numerical experiments are performed to study the shallow-water island wakes with a shelf slope. The shelf slope provides the inhomogeity of the bottom stress. The two vorticity generation mechanisms are presented in the shallow-water island wake, which is different from the deep water wake, where only the lateral stress is the source of the vorticity, and a sea mountain, where only the inhomogeneity in the bottom stress generate the vorticity. It is found that the frontal jet is formed when the stratification and bottom topography are interacted when the bottom stress is presented. The frontal jet is mainly located within the bottom boundary layer, which results in the different vertical structure of the island wakes in shallow water and deep water. The PV anomaly analysis shows the diapycnal process and frictional processes play different role in the PV anomalies generation which give rise to the asymmetry between the northern and southern sides of the island. The solution is very sensitive to the bottom stress based on the comparison between the case with and without the bottom stress. The baseline experiment solution converges to the deep-water island wake when the shallowest water depth increases. When th e island is sunken into water, i.e., a sea mountain case, the lateral stress is absent and the surface vorticity is much weaker than that with the island presented. 20

Acknowledgments CD and JCM would like to thank the support from the National Science Foundation (OCE 06-23011) and NASA project (NASA 2007). Part of works is implemented when CD visited the Dynamical Meteorological Lab / Ecole Normale Superieure /Ecole Polytecheque, Paris, France in the summer of 2008 with the financial support from the EU Sustainable Development, hosted by AS. References: Dong, C. D., J. McWilliams, and A. F. Shchepetkin, 2007: Island wakes in deep water, J. Phys. Oceanogr., 37, 962-981. 21

Perrer, G., A. Stegner, M. Farge, and T. Pichon, 2006: Cyclone-anticyclone asymmetry of large scale wakes in the laboratory, Phys. Fluids, 18, 036603, doi: 10.1063/1.2179387. Thomas, L.2007: Formation of intrathermocline eddies at ocean fronts by wind-driven destruction of potential vorticity. Tomczak, M., 1988: Island wakes in deep water and shallow water, J. Geophys. Res. 93, 5153-5154. 22