Island Wakes in Shallow Water

Changming Dong, James C. McWilliams, et al

Institute of Geophysics and Planetary Physics, University of California, Los Angeles

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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.

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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 shallow-

water 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

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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 deep-

water 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.

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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 sea-

mountain (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 ≤ r0 h (x, y) = { (1) = (hmax + hmin)/2 + (hmax-hmin)/2 * tanh ( (r – rbreak)/rwidth) , r > r0

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where (x0,y0) is the center of the island, r = [(x-x0)2+(y-y0)2]1/2 is the distant from the

center of the island, hmax and hmin are the deepest water depth in the domain, rbreak is the

distance of the shelfbreak from the center of the island, and rwidth 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 r0= 5km the shallowest water depth hmin =

50 m, the deepest water depth offshore hmax = 500m, the shelf break width is rbreak =

6

0 20 40 60 80 100 120

0

50

100

150

200

250

300

350

400

450

500

Cross−shelf Distance (km)

Dep

th (

m)

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.

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20km, and the shelf width is rwidth = 8km, see Fig. 1. The quadratic bottom stress is

applied, τ = Cd Ub |Ub|, where the Cd = 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

Cd

Island Radius With/Without

Shelf Slope

Shallowest

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

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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

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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.

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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

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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)

.

he PV equation is

(3) ional force F = (X, Y),

(4)

T

where the frict

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(5)

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).

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

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the PV ithin

e shelp slope levels and increase again below the below Level (200m).

(6)

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).

d

e

enter of the island is chosen for estimate the PV balance in the southern (northern) part

Fig. 5 the isotherm lines (18.5 and 22.5, the thick black lines) are chosen for the volume to estimate the PV balance.

increase above the shelf break level (Level 10m and 50m) but decreases w

th

The PV Eq. (3) can be rewritten as

To calculate th as (2007) is used to e PV balance, the methodology developed by L. Thom

estimate each term in Eq. (6). A water volume which is bounded by the isotherm 18.5 an

22.5, the edge island, 50km upstream and downstream, and 30km south (north) of th

c

of the island.

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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.

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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

seen in Sect 3, the bottom stress plays an important role in the vorticity

e effect of the bottom stress on the PV. According to the Ertel PV definition, it is

omposed of three terms, which are related to the vertical density gradient, y-direction

ensity gradient and x-direction density gradient:

As we have

generation. In Case 2, the bottom stress is set to be zero, i.e., slippery bottom. We discuss

th

c

d

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

eglected. It can seen that without bottom stress, the PV anomalies are barely seen on the

land, 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.

n

southern side of the is

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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

rm 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

te

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vorticity generated from the off-touched layer decrease with the water depth of the shelf

increasing. When hmin = 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 th (hmin) on Day 20: hmin=25m (Case 3), 50m (Case 1),

75m (Case 4) and 100m (Case 5). water dep

4.3 Sea Mountain

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)

an 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

weaker th

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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

experime t water

hen th sea mountain case, the

nt solution converges to the deep-water island wake when the shallowes

depth increases. W e island is sunken into water, i.e., a

lateral stress is absent and the surface vorticity is much weaker than that with the island

presented.

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Acknowledgments CD and JCM would like to thank the support from the National

cience Foundation (OCE 06-23011) and NASA project (NASA 2007). Part of works is

CD visited the Dynamical Meteorological Lab / Ecole Normale

uperieure /Ecole Polytecheque, Paris, France in the summer of 2008 with the financial

., J. McWilliams, and A. F. Shchepetkin, 2007: Island wakes in deep water, J.

Phys. Oceanogr., 37, 962-981.

S

implemented when

S

support from the EU Sustainable Development, hosted by AS.

References:

Dong, C. D

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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.

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