Nonlinear internal waves in Massachusetts Bay: Using a model to make sense of observations A. Scotti University of North Carolina

Many thanks to R. Beardsley R. Beardsley B. Butman B. Butman J. Pineda J. Pineda R. Grimshaw R. Grimshaw NSF and ONR NSF and ONR

Outline Geographical setting Geographical setting Observations Observations Modeling strategy Modeling strategy Generation/propagation Generation/propagation Shoaling Shoaling Late stage propagation Late stage propagation 3D effects 3D effects

NLIW’s Generation Interaction of barotropic flow with topography (sills, shelf/slope, canyons…) Interaction of barotropic flow with topography (sills, shelf/slope, canyons…) Nonlinear steepening of internal tide Nonlinear steepening of internal tide Collapse of mixed regions Collapse of mixed regions Baroclinic instabilities of jets Baroclinic instabilities of jets …

NLIW’s propagation Nonlinearity tends to steepen the fronts Nonlinearity tends to steepen the fronts Dispersion acts to smear the fronts Dispersion acts to smear the fronts Weakly-nonlinear theories achieve a balance between the two. Weakly-nonlinear theories achieve a balance between the two. Simplest case Korteweg-deVries equation: amplitude determines speed and wavelength Simplest case Korteweg-deVries equation: amplitude determines speed and wavelength

NLIWs shoaling/interaction with topography Scale mismatch: topographic scales O(km), NLIWs scales O(<100m). Scale mismatch: topographic scales O(km), NLIWs scales O(<100m). Is weakly nonlinear theory appropriate? Is weakly nonlinear theory appropriate? Effects of dissipation? Effects of dissipation? 3D effects? 3D effects?

Nonlinear internal waves in Massachusetts Bay

Observations in Massachusetts Bay Halpern (J.G.R 1971, J. Mar. Res. 1971) Halpern (J.G.R 1971, J. Mar. Res. 1971) Haury et al. (Nature 1979, J. Mar. Res. 1983) Haury et al. (Nature 1979, J. Mar. Res. 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 et al., JFM, 2006; JGR 2007, 2008) MBIWE98 (Scotti et al., JFM, 2006; JGR 2007, 2008)

MBIWE 98: Experiment layout

Generation Propagation as wave of depression Shoaling and conversion to elevation, along gentle shoaling area

2D Modeling approach The model solves the Euler equation in 2D along the line joining the MBIWE98 stations. The model solves the Euler equation in 2D along the line joining the MBIWE98 stations. Spectral discretization Spectral discretization Realistic topography and stratification Realistic topography and stratification Forced with barotropic tide Forced with barotropic tide Hydrostatic approximation recovered if cut-off imposed at large scales O(100 m) (Scotti and Mitran, Ocean Modeling, 2008). Hydrostatic approximation recovered if cut-off imposed at large scales O(100 m) (Scotti and Mitran, Ocean Modeling, 2008).

Generation/Propagation Effects of environmental (heaving of thermocline) and forcing parameters (spring/neap cycle)

Generation: CTD observations (Geyer and Terray, unpublished) and model Model Observation End of ebb phase

From standing wave to undular bore Beginning of flood phase

Nonlinearity and dispersion effects during generation

Evolution of the undular bore Standard conditionsSpring 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)

Energy radiated and pycnocline depth Positive correlation is found between pycnocline depth and intensity of energy radiated. Positive correlation is found between pycnocline depth and intensity of energy radiated. However, total radiation ~ 20 MW is negligible compared to barotropic tide (~20 GW) However, total radiation ~ 20 MW is negligible compared to barotropic tide (~20 GW)

Shoaling Interaction with shoaling topography Interaction 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.

Larger amplitude Smaller amplitude Modeled shoaling

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

Measured (lines) and predicted (symbols) current at the 50-m isobath. Bottom current Surface current Mid depth current. Notice the change in direction! Comparison model/observation

Nonlinearity vs. dispersion during BCEs Nonlinearity alone captures essential aspects of BCEs.

Nonlinear effects of interaction with topography with a 2-layer hydrostatic model The propagation speed of a point on the interface depends on the total depth, the thickness of the lower and upper layer and the velocity difference across the layer The propagation speed of a point on the interface depends on the total depth, the thickness of the lower and upper layer and the velocity difference across the layer Barotropic advection Buoyancy speed Total speed

Shoaling in a 2-layer hydrostatic 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 shoaling area: fully nonlinear vs. weakly nonlinear models. 2-layer, hydrostatic, fully nonlinear. Extended KdV KdV

When to expect BCEs. The undular bore cannot propagate undisturbed past the point where the total depth equals twice the displacement of the pycnocline. The undular bore cannot propagate undisturbed past the point where the total depth equals twice the displacement of the pycnocline. The shoaling bottom acts as a low-pass filter. The high- frequency content is lost to instabilities. The internal tide propagates inshore as a wave of rarefaction followed by a bore that restores the stratification. The shoaling bottom acts as a low-pass filter. The high- frequency content is lost to instabilities. The internal tide propagates inshore as a wave of rarefaction followed by a bore that restores the stratification. The energy dissipated in the process is about 35% of the flux just before the shoaling. Thus, a significant fraction of baroclinic energy is radiated inshore. The energy dissipated in the process is about 35% of the flux just before the shoaling. Thus, a significant fraction of baroclinic energy is radiated inshore.

Life after a BCE. NLIWs in the shallow 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 found in the trailing edge waves

MLIWs induced mixing Several assumptions: Several assumptions: –Unidirectional waves –Dissipation occurs along thermocline –Background state steady Mean dissipation 1.3x10 -3 W/m 3 Mean dissipation 1.3x10 -3 W/m 3 Mixing efficiency 2.5x10 -2 Mixing efficiency 2.5x10 -2 K H = 2.5x10 -5 m 2 /s K H = 2.5x10 -5 m 2 /s Heat flux across thermocline: 140 W/m 2 Heat flux across thermocline: 140 W/m 2 Net heat flux at the surface: 180 W/m 2 Net heat flux at the surface: 180 W/m 2 Barry et al., ‘Aha Huliko’a 2001 Winter workshop

Three-dimensional effects

Conclusions Nonlinearity alone captures essential aspects of physics in Massachusetts Bay during generation and shoaling. Nonlinearity alone captures essential aspects of physics in Massachusetts Bay during generation and shoaling. Bottom collision events can be predicted based on 2-layer hydrostatic models. Bottom collision events can be predicted based on 2-layer hydrostatic models. Evolution after BCEs gives rise to triangular or rectangular bores in the shallow reach. Evolution after BCEs gives rise to triangular or rectangular bores in the shallow reach. Trapped cores within waves of elevation are found sometimes in the trailing edge waves. Trapped cores within waves of elevation are found sometimes in the trailing edge waves.

Outstanding issues Composition of packets highly variable. What controls it? Composition of packets highly variable. What controls it? Energy focusing. How to model it? Energy focusing. How to model it? Effects of friction and instabilities on formation and propagation of waves with trapped cores. Effects of friction and instabilities on formation and propagation of waves with trapped cores. Mixing and transport. Mixing and transport.