ISLAND WAKES GENERATED ISLAND WAKES GENERATED BYBY

AN ELLIPTICAL TIDAL FLOWAN ELLIPTICAL TIDAL FLOW

Philippe EstradePhilippe Estrade

Jason MiddletonJason Middleton

University of New South WalesUniversity of New South Wales

Outline

➢ introduction➢ model set-up➢ some 2D results➢ some 3D results➢ concluding remarks

Introduction

• Wake flows often govern near shore environmental processes (sediments, nutrients, pollutants, ...)

• Wakes generated by constant upstream flows are well documented however oceanic flows are unsteady and non-uniform (topography, tidal currents, forcing variability, ...)

• Transient wake generated by unidirectional tidal flow have been addressed for headland (Signell & Geyer, 1991) & inner shelf island (Rattray island) but not for mid & outer shelf island for which the tidal flow is not polarized

• Impact of an elliptical tidal current on island wake structure ?

• ROMS has been used in idealized configurations to address this question ...

Dynamical context :

“isolated outer shelf island in shallow water” (e.g. Great Barrier Reef) :homogeneous fluid flowing around a topographic obstacle both in 2D & 3D

3D :

2D :also barotropic mode in 3D

Dynamicalpreconditioning :

Topography :

(circular island with or without surrounding bathymetry)

+ additional eddy viscosity within sponge layers(600 m wide - up to 10 m²/s at open boundaries)

3D :

cyl gauss

Forcing and boundaryconditions : all boundaries are open with :

2D & 3D : FS_CHAPMAN & M2_FLATHER(η,u,v specified analytically)

3D : M3_RADIATION & T_RADIATION(u,v,w,T not specified)

η v u given by basic [2D – flat bottom – linear – inviscid] theory :

solution can be written as a linear combinationof inertia gravity waves (igw) :

where are input parametersfor each tidal component of interest

wave number(s) given by the dispersion relation :

η u

η v

η u

η v

Elliptical tidal forcing :

Study limited to 1 component : meridional (kx=0, k=ky) & semi-diurnal wave (period T=12h) ==> 2 cases :

Progressive wave : Standing wavesingle wave (k) incident (k) + reflected wave (-k)

similar elliptical forcing can be found with adequate choice of ysw & tsw

η0 (m) latitude (°N)

e1 0.5 15

e2 0.455 30

e3 0.265 60

flood/ebbin phase vs out of phase

high/low

a “control” run :2D / flat bottom (no island) / e1 grid : 200*300 ; res=50 m ; dt=1s

“error” < 5% except : when t/T ~ n+1/4 or n+3/4 ( η & v ~ 0 )

when t/T ~ n or n +1/2 ( u ~ 0 ) the numerical solution propagates almost like the linear igw (used to force the model at the 4 open boundaries) despite non-linearity (advection, bottom friction)

12 experiments :

Spin-up :(insignificant difference

between 2D/3D& cyl/gauss)

grid :2D cyl e1 400*450

/ / e2 450*5503D gauss e3 600*600

2D & 3D : horizontal resolution 50 m barotropic mode 1 s

3D : baroclinic mode 48 s 20 σ levels

initial conditions : from rest

some 2D results (vorticity & circulation) :

thin ellipse (e1 & e2) : transient 2 eddies structuresduring flood & ebb phases & dissipation in between(stronger activity with varying bottom topography)

large ellipse : vorticity filamentscontinuously progressing

(weak sensitivity to topography)

about the free surface elevation (2d/gauss/e1) :

<η> :

η-ηpw :

residual :

Larger negative “anomalies” in the flow separation regions leads to residual depression along minor axis despite residual flow convergence

residual circulationand vorticity (2D & 3D) :

the tidal current rotation favors the development of the eddy rotating in the

same direction and weakens the development of the second eddy

no qualitative difference in η, u, v between 2D & 3D modelling

(for this particular range of parameters !)

some 3D results (vertical velocity) :

z = -H/2

Upwellings are : stronger with a varying topography weaker with larger ellipse

Which mechanisms drive these vertical motions ?

Secondary circulation & vertical motions :

Alaee et al (2004) : flow curvature can generate significant vertical motions by convergence/divergence of the secondary circulation

(u’,v’)t/T=5.1

3D/gauss/e1

vertically integrating the continuity equation from the bottom (or from the surface) to depth z and then replacing (u,v) by (u’,v’) + (u,v) gives w = wp + ws :

Example vertical velocity decomposition :

t/T = 5.1z = -3/4.H

Vertical motions mainly stem from :

flow curvature (i.e convergence/divergence of the secondary circulation) for a cylindrical island

Combination of wp & ws

for a varying topography

Residual vertical velocities :

z = -3/4.H

3D/gauss/e1 & e2 :

wide & strongresidual upwelling

along the major axis

consistent feature ?

Concluding remarks :

➢ further sensitivity studies needed to identify wake regimes vs relevant dimensionless numbers (h0/[Cd.R], V0/[ω.R], ω/f, others ?)

➢ similar tidal forcing method can be applied for headland or innershelf island wake studies (Kelvin waves propagating along a closed boundary) but limited to idealized configurations as well (flat bottom near open boundaries)

➢ study motivated by transient wake observations around LEI (Lady Elliot Island – Great Barrier Reef)

➢ current studies : adding missing LEI ingredients (stratification, buoyancy flux, wetting & drying, wind, neap/spring M2 & S2, ...) and applying GST tools

➢ heading for realistic modelling of LEI ...