Internal Tide Generation Over a Internal Tide Generation Over a Continental ShelfContinental Shelf

Summer 2008 internshipSummer 2008 internship

GaGaёёlle Faivrelle FaivreFlavien Gouillon, Alexandra BozecFlavien Gouillon, Alexandra Bozec

Pr. Eric P. ChassignetPr. Eric P. Chassignet

BiographyBiography

Gaëlle FaivreGaëlle Faivre

Student from the Engineering Student from the Engineering School in Mathematic Modeling School in Mathematic Modeling and Mechanics (MATMECA) and Mechanics (MATMECA)

Bordeaux, FRANCEBordeaux, FRANCE

Position at COAPS:Position at COAPS:

Internship from June 2 - Sept Internship from June 2 - Sept 16, 200816, 2008

OutlineOutline

I. Introduction

II. Motivation

III. Objective

IV. Approach

- Analytical solution

- Numerical experiment

V. Results

VI. Conclusion

I. IntroductionI. Introduction

Internal waves review:Internal waves review:

- Internal waves occur in stably stratified fluidsInternal waves occur in stably stratified fluids when a water when a water parcel is displaced by some external force and is restored parcel is displaced by some external force and is restored by buoyancy forces. Then the restoration motion may by buoyancy forces. Then the restoration motion may overshoot the equilibrium position and set up an oscillation overshoot the equilibrium position and set up an oscillation thereby forming an internal wave.thereby forming an internal wave.

- Internal tide comes from the Internal tide comes from the interaction between rough interaction between rough topography and the barotropic tidetopography and the barotropic tide..

- Important internal wave generation occurs at the Important internal wave generation occurs at the shelf breakshelf break where the slope is where the slope is abruptabrupt..

- Horizontal length scales of the order 1 to 100 km- Horizontal length scales of the order 1 to 100 km

- Horizontal Velocity of 0.05 to 0.5 m.s- Horizontal Velocity of 0.05 to 0.5 m.s-1-1

- Time scale of minutes to days- Time scale of minutes to days

Surface signature of an internal wave Surface signature of an internal wave

An example of a surface signature of an internal waveAn example of a surface signature of an internal waveView from a boatView from a boat

II. MotivationII. Motivation

- Knowledge of internal wave generation and how they Knowledge of internal wave generation and how they propagate is crucial to understand ocean mixing and the propagate is crucial to understand ocean mixing and the large scale ocean circulation.large scale ocean circulation.

- Internal waves generation at the shelf affects:Internal waves generation at the shelf affects:- sediment transportsediment transport- biologybiology- oil production companiesoil production companies- submarine navigation.submarine navigation.

- At the shelf, the dynamic of internal wave is At the shelf, the dynamic of internal wave is strongly strongly non-hydrostaticnon-hydrostatic and thus cannot be well resolved in and thus cannot be well resolved in Oceanic General Circulation Models that usually make Oceanic General Circulation Models that usually make the hydrostatic approximationthe hydrostatic approximation

III. ObjectivesIII. Objectives

- Assess the HYbrid Coordinate Ocean Model Assess the HYbrid Coordinate Ocean Model (HYCOM) skills to simulate internal wave at an (HYCOM) skills to simulate internal wave at an abrupt slope.abrupt slope.

- Evaluate and document the limitation of Evaluate and document the limitation of HYCOM on representing these waves.HYCOM on representing these waves.

IV. ApproachIV. Approach

1.1. Compute an analytical solution for an Compute an analytical solution for an idealized case of internal tide generation over idealized case of internal tide generation over a shelf breaka shelf break

2.2. Build the same configuration with HYCOMBuild the same configuration with HYCOM

3.3. Compare the dynamical properties as well as Compare the dynamical properties as well as the energetic of the generated internal wave the energetic of the generated internal wave for both results.for both results.

Analytical solutionAnalytical solution

Based on Based on Griffiths and GrimshawGriffiths and Grimshaw, (2004) , (2004)

Lh

Conditions for this analytic results:Conditions for this analytic results: - The flow is two-dimensional - The flow is two-dimensional - 2 layers ocean- 2 layers ocean - The interface between the 2 - The interface between the 2 layers needs to be smaller thanlayers needs to be smaller than the shelf depththe shelf depth

There is one There is one baroclinic mode with baroclinic mode with the phase speed:the phase speed:

)(1 LL hhcc

)(1 RR hhcc

Wave speed at the shelf Wave speed at the shelf

Wave speed in the deep ocean :Wave speed in the deep ocean :

Figure 1: schematic of the analytical modelFigure 1: schematic of the analytical model

Dimensions :

kmLL SC 100

mhL 200mhR 2000

2/12/1

01 1

h

dgdc

),0,( tzxz

• Low energy fluxes located each , where the slope accommodates an integer number of wavelengths of the internal wave

Analytical energy computationAnalytical energy computation

LJ RJ

L

R

x

x c

c

sdxxk

R

L

log1

)(1

Sf

LR

L

ccs

1

22 ff

Prediction of the low energy flux

The nondimensional depth-integrated energy fluxes at the shelf are given by:The nondimensional depth-integrated energy fluxes at the shelf are given by:

)1()1(22

22

Cf

R

R

L

R

LL L

c

c

c

c

cJ

Semi-diurnal frequency:

Steepness Steepness parameter:parameter:

)12( n

• Change in phase across the slope is given by:

Pulsation:14104.1 s

Frequency: 4101 f 1s

JJLL

• High energy fluxes located each

n2

HYCOMHYCOM

• HYCOM is run in fully isopycnal mode for this configuration.HYCOM is run in fully isopycnal mode for this configuration.

• We used all the same parameters as the analytical configuration and vary the We used all the same parameters as the analytical configuration and vary the steepness parameter by changing the stratification, and the interface depth but steepness parameter by changing the stratification, and the interface depth but keeping a constant shelf length/coast length ratio (keeping a constant shelf length/coast length ratio (LLss/L/Lcc).).

• Objectives: Show that low energy fluxes are simulated in HYCOM and well Objectives: Show that low energy fluxes are simulated in HYCOM and well located when located when LLss/L/Lcc=1=1..

Schematic of the Model Configuration Analytical Energy Fluxes

Energy fluxes as a function of the steepness parameter (function of the wave speed at the shelf) for

Low energy fluxes Low energy fluxes expected atexpected at

Results: Energy fluxes comparisonResults: Energy fluxes comparison

n2

En

erg

y fl

uxe

s (W

.m-2)

1s

1.0R

L

h

h

HYCOMHYCOM

Analytical SolutionAnalytical Solution

DiscussionDiscussion

• We obtain low energy fluxes just like the analytical solution We obtain low energy fluxes just like the analytical solution predicted. However some of them seem to be not located predicted. However some of them seem to be not located correctly for a particular choice of parameters.correctly for a particular choice of parameters.

• What could cause this shift in the low energy location?What could cause this shift in the low energy location?

– Not enough sampling (because each point is a different Not enough sampling (because each point is a different configuration)configuration)

– Several approximations and truncations are made in the Several approximations and truncations are made in the internal wave group speed and in the internal wave energy internal wave group speed and in the internal wave energy computationscomputations

– HYCOM does not represent the wave propagation correctly HYCOM does not represent the wave propagation correctly (wavelength, wave speed)(wavelength, wave speed)

ConclusionConclusion

1.1. We have found an analytical solution for the internal wave We have found an analytical solution for the internal wave generation at a shelf with a 2-layers ocean idealized problem.generation at a shelf with a 2-layers ocean idealized problem.

2.2. We have conducted numerical simulations with HYCOM for We have conducted numerical simulations with HYCOM for the same configuration.the same configuration.

• Low energy fluxes are represented in HYCOM.Low energy fluxes are represented in HYCOM.

3.3. For a particular choice of parameters, these some of the low For a particular choice of parameters, these some of the low energy fluxes seems to be shifted from where the analytical energy fluxes seems to be shifted from where the analytical solution predicts.solution predicts.

• This could be due to our mistakes or HYCOM errors on This could be due to our mistakes or HYCOM errors on the representing the propagation of the wavesthe representing the propagation of the waves

Future WorkFuture Work

• More simulations should be run in order to analyze the origin of More simulations should be run in order to analyze the origin of the shift in the low energy fluxesthe shift in the low energy fluxes

From the small scale of sFrom the small scale of s11??

Computational errors in model?Computational errors in model? Human errors?Human errors?

• Further analyses of the approximations in our computations Further analyses of the approximations in our computations should be madeshould be made

Additional model configurationsAdditional model configurations

Observe climatological density stratifications and adapt it to Observe climatological density stratifications and adapt it to realistic continental shelves.realistic continental shelves.

Examine the effects of alongshore variability in shelf Examine the effects of alongshore variability in shelf topography (application to 3 dimensions)topography (application to 3 dimensions)

Include nonlinear and nonhydrostatic termsInclude nonlinear and nonhydrostatic terms

Interface displacement of the internal wavesInterface displacement of the internal waves

5.0Lh

d

Fig2: Displacement of the interface for a horizontal normalization

Fig3: Internal tide of two-layer fluid, with ,

8.0R

L

c

c

5.0Lh

d at )2/()/log(1 LR ccs

Rh

z

Baroclinic zonal velocitiesBaroclinic zonal velocities

The interface displacement is about 1 meter and compares well The interface displacement is about 1 meter and compares well with the analytical prediction (difficult to see displacement from with the analytical prediction (difficult to see displacement from the scale of the figure) the scale of the figure)

Vel

ocity

(m

s-1)

Snapshot after 45 hours, (right after spin-up)Snapshot after 45 hours, (right after spin-up)