Non-hydrostatic effects on internal waves and mixing in the coastal ocean Jiuxing Xing and Alan M....

Non-hydrostatic effects on internal waves and mixing in

the coastal ocean

Jiuxing Xing and Alan M. Davies

(Proudman Oceanographic Laboratory, Liverpool)

Jonsmod 2006-Plymouth

Motivation

Understand small scale processes (e.g. solitary waves, convection)

Stratified (tidal) flows over the steep topography (e.g., lee waves, flow separation and eddies)

Are current coastal ocean models sufficient (e.g., is non-hydrostatic dynamics important)?

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Examples of small scale processes: ISWs of elevation on the Oregon shelf

Klymak and Moum (2003)

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ISWs in the Faeroe-Shetland Channel

Hosegood et al (2004)

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Stratified tidal flow over sills (Loch Etive, Inall et al, 2004)

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Models σ-following coordinate models (e.g.,

POLCOMS, POM, BOM) Z-coordinate models (e.g., MITgcm) The iterative method for non-hydrostatic

pressure:

• an elliptic equation for the non-hydrostatic pressure

0

0),,(),,(),(),(

zNH

tzxPzdtzxgtxgtxP

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Tests of the MIT model using Lab. exp. (Internal solitary waves over a slope (Michallet and Ivey 1999)

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Internal solitary waves (model results)

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Internal solitary waves (lab experiments vs model results)

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A dispersive ISW (small-amplitude)

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Large amplitude ISWs on a slope

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Tidal flow over a sill – lee wave generation and non-hydrostatic effects

Idealized model setup• M2 tide forced at the seaward boundary

• Closed landward boundary

• Resolution: dx=10 to 100m, dz=1m

• Minimum viscosity (Av=10-3 m2s-1, Ah=10-1 m2s-1, no explicit diffusivity)

• Initial zero velocity, N=0.01 s-1

Model domain

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Interaction of tidal waves with bottom topography: key non-dimensional parameters

The key physical parameters:

• U0, ω0, f, N, h0, L, H.

Non-dimensional parameters:

• U0 /(ω0L) the tidal excursion parameter;

• h0/H the relative height of the topography;

• [(ω02 - f 2 )/(N2- ω0

2)]1/2 the internal wave ray slope;

• h0 /L the topographic slope;

• U0 /(Nh0) the Froude number (or h’=Nh0 /U0);

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Snapshots of (T,u,w) at 23 mins (4,5,6,7/32 Tm2,non-hydro run)

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Snapshots of (T,u,w) at 23 mins (4,5,6,7/32 Tm2,non-hydro run)

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Snapshots of Ri number at 4,5,6,7/32T (non-hydrostatic run)

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Snapshots of temp & velocity at 4,5,6,7/32T (hydrostatic run)

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Temp & velocity at t=2/8T, 3/8T, 4/8T, 5/8T (non-hydrostatic run)

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Temp & velocity at t=2/8T, 3/8T, 4/8T, 5/8T (hydrostatic run)

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Vertical averaged internal wave energy flux (non-hydrostatic (left) and hydrostatic (right) )

01d

pdzuH

F

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Non-hydrostatic (top) and hydrostatic pressure

PPhh and P and Pnhnh have a 180 have a 180oo phase shift phase shift

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In a linear system, Ph & Pnh out phase

,00z

pb

z

p

t

wnh

,02

wNtb

)(nhhppp

)/(0

gb

,0

bz

ph

,0

bz

ph

0)( 2

2

2

z

pN

z

p

tnhh

)cos(

),cos(

2

2

tANz

p

thentAz

pif

nh

h

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indirect estimate of hydrostatic pressure by matching isopycnals to streamlines

nonhydrostatic pressure

seafloor value

seafloor value

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Power spectral density (w) at two locations, non-hydrostatic (left) vs hydrostatic (right) (N=0.01)

At lower frequency, as predicted by Khatiwala (2003), but not higher frequency.

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Power spectral density (w) for a steeper topography (N=0.01), non-hydrostatic (left) vs hydrostatic (right) (N=0.01)

Significant departure from recent theory at both lower & higher frequency.

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Summary

Including non-hydrostatic dynamics in the coastal ocean models is feasible.

Model results are encouraging comparing to the laboratory data.

Importance of non-hydrostatic dynamics to lee wave generation and breaking;

Strong kinetic energy spectral peak at higher (lee wave) frequency near the sill - a challenge to observationlists;

Enhanced mixing due to the smaller-scale ripple topography - a challenge to modellers;

More work is needed to assess the model quantitatively and quantify wave drag effects on mixing and circulation (3D effects may be important) .