# Presentation: 942140

## Tidal mixing at steep topography

- Sonya Legg
- Princeton University, NOAA-GFDL

## Tidal Energy Budget

- Munk and Wunsch, 1998

- Most tidal energy is dissipated in coastal oceans, but the small amount dissipated in deep ocean has large impact on climate.
- Different climate scenarios (e.g. raising/lowering sealevel) would have different dissipation patterns.

## Mechanisms of tidal mixing in deep ocean

- 1/3 of energy from ocean tides is dissipated in deep ocean.
-  Used for mixing stratified ocean interior.
- Some mixing local to topography, e.g. mid-ocean ridges, seamounts
- Some energy carried throughout ocean by waves, leading to distributed mixing.

- Tidal energy flow chart.

- Unknowns
- How much energy is extracted from tides?
- How is it initially partitioned between waves and mixing?
- Where do waves eventually break and cause mixing?

## Where does tidal mixing happen? 

- Observations (*Polzin et al, 1997*) show interior mixing is concentrated over rough topography, e.g. mid-ocean ridges and seamounts

## Evidence for tidal mixing over a knife-edged ridge: the Hawaiian Ocean Mixing Experiment (Klymak et al, 2005)

- Diffusivity (estimated from measured dissipation) enhanced over ridge

- Dissipation scales with M2 tidal energy flux (Klymak et al, 2005)

## Governing parameters for tidal flow over topography

- Topography: height  *h*, width *L*, depth* H*

- Flow: speed *U*, oscillation frequency 

- Nondimensional parameters

- Topography

- Flow

- Others: coriolis *f*, buoyancy frequency *N*

- Wave slope

- Relative steepness

- Relative height

- Tidal excursion

- Froude numbers

- Analytical studies assume some or all of topographic/flow parameters are small – numerical simulations don’t have this restriction.

*H*

*L*

*h*

## Internal tide generation by finite-amplitude barotropic tide

- Early theoretical predictions (e.g. Bell 1974) assume gentle, low amplitude topography ( h/H << 1).
- Recent numerical simulations (e.g. Khatiwala, 2003) and theory (e.g. Balmforth et al, 2002; St Laurent et al, 2003; Llewellyn Smith and Young, 2003) examine how tidal conversion depends on finite steepness and  relative height h/H, for small amplitude flow.
- As                    energy  conversion doubles in deep fluid.
- As h/H          1, energy conversion is further increased.

- Q: What happens in nonlinear regimes when RL > 1, and Fr = U/(Nh) < 1?

- Khatiwala 2003

- Gaussian topography

- St Laurent et al, 2003
- Knife-edge topography

- Increasing h/H

- How much energy is extracted from the barotropic tide?

- Increasing 

## Numerical simulations of finite amplitude tidal flow over Gaussian topography. 

- Numerical simulations of finite amplitude tidal flow over Gaussian topography.

- Baroclinic velocity snapshots from simulations of tidal flow over Gaussian topo with forcing amplitude U0=2cm/s (Legg and Huijts, 2006; using MITgcm)*.*

- Low, wide, shallow topo, 

- Low, narrow, steep topo, =2.0

- Tall, steep topo, 

- Steep topography leads to generation of internal tide beams: energy concentrated on wave characteristics.

- Key questions
- Do theoretical predictions hold for large amplitude flows?
- How much of converted energy is dissipated locally v. radiated away?

## Generation of higher harmonics at narrow topography

- Baroclinic velocity snapshots for low, narrow, steep topography, 

- U0=8cm/s

- U0=16cm/s

- U0=24cm/s

- More beams, at steeper angles, corresponding to 2 appear as U0 increases.

- Actual power spectra

- Theoretical power spectra

- Bell’s theory predicts higher harmonics well

- RL = 1.4

- RL = 2.8

- RL = 4.2

## Isopycnal deflection by large amplitude tides: U0=24cm/s

- Low wide shallow topo, RL<<1, Fr=1.5

- Low narrow steep topo, RL = 4.2, Fr=1.6

- Tall steep topo, RL <1, Fr=0.1

- Tall steep narrow topo, RL = 1.5, Fr=0.1

- Large amplitude forcing over large amplitude steep topo leads to local overturns in internal hydraulic jump-like features

## Quantitative results: energy conversion

- For wider topo, only 10% of energy extracted from tide is dissipated locally; for narrow topo, much greater fraction.

- Rate of energy conversion from barotropic tide

- Ratio of dissipation rate to conversion rate

- Low, wide topography; low, narrow topography; tall wide topography; tall narrow topography

- Bell’s prediction

- St Laurent et al prediction for steep topo, deep fluid

- St Laurent et al prediction for steep topo, h/H=0.5

- Theoretical predictions of energy conversion agree well with numerical model results.

- All from Legg and Huijts, 2006; using MITgcm

## Focus on hydraulic jump-like features Realistic tall steep topography:  Hawaiian ridge

- 46km

- -700m

- -1760m

- Hawaiian ridge is tall and steep.
- > 1, RL < 1, Fr << 1, h/H -> 1.
- Hydraulic jumps develop over steep slope during downslope flow: at flow reversal, jumps propagate upslope as internal bores.

- Buoyancy field for U0=5cm/s, M2 tidal forcing.

- Asymmetric response: details of slope are important.

- Stratification and topography data from Kaena ridge courtesy of Jody Klymak and HOME researchers.

## Role of topographic slope in generating transient jumps

- i.e.

- So transient jump-like waves may be possible if slope is sufficiently steep that Fr < 1

- In steady state flows, strongly nonlinear leewaves are generated when

- Fr << 1: flow is blocked by topography.
- Fr >> 1: small amplitude linear leewaves.

- By analogy with steady case, we expect strongly nonlinear waves if h is large enough that

## Evidence for importance of Fr: 

-  isopycnal displacement amplitude h

- h v. vertical tidal excursion dh/dx U0/

- h v. U0/N

- h v. dh/dx U0/U0/N/0.3

- At large Fr, h is governed by tidal excursion distance, but at small Fr, h is proportional to U0/N, the limit based on KE to PE conversion. There is therefore a regime transition governed by Fr.

## Understanding the overturning at the Hawaiian ridge: numerical experiments

- Analysis suggests that the essential ingredients for the jump-like features with associated overturning are (a) large amplitude flow over large amplitude topography (b) steep topographic slope.

- Simulation with idealized topography and uniform stratification. The behavior seen in the realistic simulation is reproduced when stratification at shelf-break and maximum slope are the same.

##  Dependence of overturning on slope

- Real topography, Fr

- dh/dx(max) = 0.2, Fr

- dh/dx(max) = 0.1, Fr=0.6

- dh/dx(max) = 0.06, Fr

- Borelike features are found only for Fr<<1.

- Snapshots of U(color) and buoyancy (contours) just after flow reversal, all with U0=5cm/s

## Dependence of overturning on flow amplitude

- U0=2cm/s, c=186m

- U0=5cm/s, c=465m

- U0=10cm/s, c=930m

- Snapshots of U (color) and buoyancy (contours) just after flow reversal, for dh/dx(max) = 0.2. (Fr=0.3)

- Larger amplitude flow increases vertical extent and vigor of overturning.

## Evidence for overturning and internal bores

- Height of Ri<1 layer at topographic depth of -1150m

- Skewness of dT/dt at top of slope, h=-1028m.

- Steeper slopes and stronger flows associated with deeper overturning layer, occurring just after flow reversal.

- Negative dT/dt skewness for all low Fr runs indicates sudden drops in temperature, associated with upslope propagating internal bores.

## Gerneration of internal bores by reflection of internal waves on near-critical slope

- Numerical simulations demonstrate mixing is possible at all types of critical slopes. (*Legg and Adcroft, 2003*).

- 3km

- 200m

- Wave breaking is induced by reflection from near-critical slope, i.e. when .

- Buoyancy field for 1st mode internal tide

- In Hawaii scenario, internal wave impinging on near-critical slope is locally generated - relaxing internal hydraulic jump.

## Influence of internal bores on dissipation

- The region affected by internal bores has an order of magnitude higher dissipation than the internal wave beams.

- Log10 dissipation (time-averaged) for U0=5cm/s, dh/dx=0.2

- Log10 dissipation (time-averaged) for U0=5cm/s, realistic topography and stratification

## Impact of different slopes, tidal velocities, on dissipation near shelf-break

- Steep slopes and large amplitude flows have greatest enhancement of  dissipation on upper slope. High frequency flows are complicated by Parametric Subharmonic Instability.

- Ratio between upper slope dissipation and total dissipation

## Do tidally driven transient hydraulic jumps matter on a global scale? 

- Do tidally driven transient hydraulic jumps matter on a global scale?

- Fr (N/( dh/dx)) calculated on 1⁄4 degree scale

- Amplitude of tidal velocity projected onto direction of topographic gradient (cm/s)

- Large velocities combined with Frmay lead to local overturning in jump-like features: seen in several interesting locations....

## Tidal energy flow chart revisited....

- Wave-wave
- interactions

- Wave-topography
- interactions

- Barotropic tides

- Rough topography

- Internal tides

- Local
- mixing

- Wave steepening
- and breaking

- Remote and
- local mixing

**90%**

**10%**

- For slopes within a range of critical angle

- Simulations have shown where internal tides will break and cause mixing – on slopes with angles within range of critical angle.

- Simulations show that only 10% of energy is dissipated locally, except for very narrow topography.

- Dissipation is located in beams and at near-critical slopes.

## Parameterizing tidal mixing in ocean models

- St Laurent et al, 2002, implemented in GFDL MOM by Simmons et al, 2004

- Spatially variable diffusivity:

- Where: q = fraction of energy dissipated locally – set to 1/3.
- This should be a function of the horizontal length-scales of the topography

-  = mixing efficiency – set to 0.2

- E(x,y) = energy extracted from barotropic tide:

- F(z) = vertical structure function:

- = vertical decay scale – set to 500m.

-  = constant background diffusivity, accounting for remote mixing: Need to account for spatial variations in remote mixing, e.g. internal tide breaking.

- Does not account for preferred locations of mixing: e.g. beams, steep slopes.

## Conclusions

- For steep slopes and large amplitude flows, overturning internal bores are generated.
- The internal bores are locations of enhanced dissipation.
- Probable cause: transient jump-like nonlinear lee-waves are generated during off-slope flow, propagate toward slope when flow relaxes, and then undergo reflection from slope, causing overturning.
- Nonlinear jump-like features only appear when Fr based on tidal excursion distance is small, implying steep slopes.
- Vertical scale of nonlinear features depends on flow amplitude, stratification.