Nonlinear internal waves in Nonlinear internal waves in Massachusetts Bay: Using a Massachusetts Bay: Using a

model to make sense of model to make sense of observationsobservations

A. ScottiA. Scotti

University of North CarolinaUniversity of North Carolina

Many thanks toMany thanks to

• R. BeardsleyR. Beardsley

• B. ButmanB. Butman

• J. PinedaJ. Pineda

• R. GrimshawR. Grimshaw

• NSF and ONRNSF and ONR

OutlineOutline

• Geographical settingGeographical setting

• ObservationsObservations

• Modeling strategyModeling strategy

• Generation/propagationGeneration/propagation

• Shoaling Shoaling

• Late stage propagationLate stage propagation

• 3D effects3D effects

Nonlinear internal waves in Nonlinear internal waves in Massachusetts BayMassachusetts Bay

Observations in Massachusetts Observations in Massachusetts BayBay

• Halpern (J.G.R 1971, J. Mar. Res. 1971)Halpern (J.G.R 1971, J. Mar. Res. 1971)• Haury Haury et alet al. (Nature 1979, J. Mar. Res. . (Nature 1979, J. Mar. Res.

1983)1983)• Trask and Briscoe (J.G.R. 1983)Trask and Briscoe (J.G.R. 1983)• Chereskin (J.G.R. 1983)Chereskin (J.G.R. 1983)• Scotti and Pineda (GRL, 2004)Scotti and Pineda (GRL, 2004)• MBIWE98 (Scotti MBIWE98 (Scotti et alet al., JFM, 2006; JGR ., JFM, 2006; JGR

2007, 2008)2007, 2008)

MBIWE 98: MBIWE 98: Experiment Experiment

layoutlayout

GenerationPropagation as wave of depression

Shoaling and

conversion to

elevation, along gentle

shoaling area

2D Modeling approach2D Modeling approach• The model solves the Euler The model solves the Euler

equation in 2D along the line equation in 2D along the line joining the MBIWE98 stations. joining the MBIWE98 stations.

• Spectral discretizationSpectral discretization

• Realistic topography and Realistic topography and stratification stratification

• Forced with barotropic tideForced with barotropic tide

• Hydrostatic approximation Hydrostatic approximation recovered if cut-off imposed at recovered if cut-off imposed at large scales O(100 m) (Scotti and large scales O(100 m) (Scotti and Mitran, Ocean Modeling, 2008).Mitran, Ocean Modeling, 2008).

Generation/PropagationGeneration/Propagation

Effects of environmental Effects of environmental (heaving of thermocline) and (heaving of thermocline) and

forcing parameters forcing parameters (spring/neap cycle)(spring/neap cycle)

Generation: CTD observations (Geyer Generation: CTD observations (Geyer and Terray, unpublished) and modeland Terray, unpublished) and model

Model

Observation

End of ebb phase

From standing wave to undular From standing wave to undular borebore

Beginning of flood phase

Nonlinearity and dispersion Nonlinearity and dispersion effects during generation effects during generation

Evolution of the undular Evolution of the undular borebore

Standard conditions Spring tide

The model predicts the formation of the undular bore. However, the high-frequency oscillations develop more slowly than observed. Note that rank-ordering not always observed

Sample of T record 5 km west of SB (A)Sample of T record 5 km west of SB (A)

ShoalingShoaling

• Interaction with shoaling topographyInteraction with shoaling topography

• Bottom Collision Events (BCEs)Bottom Collision Events (BCEs)

Undular bores at the 45 m isobath: Examples from observations.

Still well offshore of location where coefficient of KdV quadratic vanishes.

Largeramplitude

Smalleramplitude

Modeled shoalingModeled shoaling

Modeled temperature field at different depths along the shoaling region =>Eulerian measurements taken at different depts show markedly different time series.

Nonlinearity vs. dispersion Nonlinearity vs. dispersion during BCEs during BCEs

Nonlinearity alone captures essential aspects of BCEs.

Nonlinear effects of Nonlinear effects of interaction with topography interaction with topography

with a with a 2-layer hydrostatic model2-layer hydrostatic model

• The propagation speed of a point on The propagation speed of a point on the interface depends on the total the interface depends on the total depth, the thickness of the lower and depth, the thickness of the lower and upper layer and the velocity upper layer and the velocity difference across the layerdifference across the layer

Barotropic advection

Buoyancy speed

Total speed

)()(

)( 212

12

xhD

dd

hD

vg

xhD

ddvc

Shoaling in a 2-layer Shoaling in a 2-layer hydrostatic modelhydrostatic model

In deep water the non linear speed is maximum at the trough thus nonlinearity steepens the front.

Past a critical depth, the maximum in c shifts towards the front of the wave, nonlinearity steepens the back, while at the same time the front becomes parallel to the bottom. Water is forced downward along the topography and the flow becomes supercritical. Instabilities develop on the back side.

Speed along inshore-moving characteristics

Characteristics along the Characteristics along the shoaling area: fully nonlinear shoaling area: fully nonlinear vs. weakly nonlinear models.vs. weakly nonlinear models.

2-layer, hydrostatic, fully nonlinear.

Extended KdV

KdV

When to expect BCEs.When to expect BCEs.

• The undular bore cannot propagate undisturbed past The undular bore cannot propagate undisturbed past the point where the total depth equals twice the the point where the total depth equals twice the displacement of the pycnocline.displacement of the pycnocline.

• The shoaling bottom acts as a low-pass filter. The The shoaling bottom acts as a low-pass filter. The high-frequency content is lost to instabilities. The high-frequency content is lost to instabilities. The internal tide propagates inshore as a wave of internal tide propagates inshore as a wave of rarefaction followed by a bore that restores the rarefaction followed by a bore that restores the stratification.stratification.

• The energy dissipated in the process is about 35% of The energy dissipated in the process is about 35% of the flux just before the shoaling. Thus, a significant the flux just before the shoaling. Thus, a significant fraction of baroclinic energy is radiated inshore.fraction of baroclinic energy is radiated inshore.

Life after a BCE. NLIWs in the Life after a BCE. NLIWs in the shallow end of Mass Bayshallow end of Mass Bay

•The model indicates that waves reorganize after a BCE.

•Possible outcomes include “squared” and “triangular” waves.

•Depth of pycnocline in shallow end determine outcome:

•if still closer to surface, “square” bores.

•if close to middepth, “triangular” bores.

“Square bore” “Triangular bore”

Observations of NLIWs in the shallow reach of Massachusetts Bay (depth 25 m).

Trapped cores are sometimes Trapped cores are sometimes found in the trailing edge found in the trailing edge

waveswaves

Three-dimensional effectsThree-dimensional effects

ConclusionsConclusions

• Nonlinearity alone captures essential aspects Nonlinearity alone captures essential aspects of physics in Massachusetts Bay during of physics in Massachusetts Bay during generation and shoaling.generation and shoaling.

• Bottom collision events can be predicted Bottom collision events can be predicted based on 2-layer hydrostatic models.based on 2-layer hydrostatic models.

• Evolution after BCEs gives rise to triangular Evolution after BCEs gives rise to triangular or rectangular bores in the shallow reach.or rectangular bores in the shallow reach.

• Trapped cores within waves of elevation are Trapped cores within waves of elevation are found sometimes in the trailing edge waves.found sometimes in the trailing edge waves.

Outstanding issuesOutstanding issues

• Composition of packets highly Composition of packets highly variable. What controls it?variable. What controls it?

• Energy focusing. How to model it?Energy focusing. How to model it?

• Effects of friction and instabilities on Effects of friction and instabilities on formation and propagation of waves formation and propagation of waves with trapped cores.with trapped cores.

• Mixing and transport.Mixing and transport.