Making practical progress in parameterizing turbulent mixing

in the deep ocean

Sonya Legg

Princeton University, NOAA-GFDL

The role of deep mixing in the general circulation

Eq Pole

z

Cooling

UpwellingOverturning

convection

overflows

tidal mixingDiapycnal mixing is necessary to close the thermohaline circulation: tides and winds are the likely source of energy for deep diapycnal mixing.

Climate models need physically based parameterizations of spatially and temporally varying tidal mixing: here we will focus on tidal mixing.

Munk and Wunsch, 1998Most 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.

Tidal Energy Budget

Wave-wave interactions

Wave-topographyinteractions

Barotropic tides

Rough topography

Internal tidesLocal mixing

Wave steepening and breaking

Remote and local mixing

•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?

Mechanisms of tidal mixing in deep ocean

Global climate models do not simulate any part of this chain of events, not just the final 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 HFlow: speed U, oscillation frequency

Nondimensional parameters

Topography

Flow

s

h

2/1

22

22

ω

ω

N

f

m

ks

Others: coriolis f, buoyancy frequency N

H

h

ωL

URL

Nh

UFr

Wave slope

Relative steepness

Relative height

Tidal excursion

Froude numbers

pw C

UFr

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

H

Lh

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. St Laurent et al, 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 when RL > 1, and Fr = U/(Nh) < 1?

Khatiwala 2003

Gaussian topography

St Laurent et al, 2003Knife-edge topography

Increasing h/H

The relevant question for mixing parameterization purposes: how much energy is extracted from the barotropic tide?

Increasing

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 topoTall, steep topo

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

Key questions for parameterization development: 1. Do theoretical predictions hold for large amplitude flows?2. How much of converted energy is dissipated locally v. radiated away?

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

Probable cause of higher relative dissipation for narrowest topography: smaller vertical

lengthscales in internal tideLow wide shallow topo Low narrow steep topo

Tall steep topo Tall steep narrow topo

Narrowest topo is the only case without energy peak at lowest vertical mode.

(Legg and Huijts, 2006)

Observations of dependence of dissipation on topographic lengthscale

Mid Atlantic ridge has much less total internal tide energy flux than Hawaii, but similar levels at high mode numbers (m>10). Dissipation levels, especially at depth, are similar, suggesting dissipation is a function of energy in high modes.

St Laurent and Nash, 2004.

Are narrow beams the only location for dissipation/mixing?

Low narrow steep topo Tall steep topo

Large amplitude forcing over large amplitude steep topo leads to local overturns in internal hydraulic jump-like features. (Legg and Huijts, 2006)

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

Q: Can internal hydraulic jumps be important at more moderate (i.e. realistic) forcing velocities?

Dissipation is all in narrow beams, no hydraulic effects

Possible transient internal hydraulic jumps are a location for overturning

Example of internal hydraulic jumps with realistic forcing, topography: Hawaiian ridge

46km

-700m

-1760m

Hawaiian ridge is tall and steep.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: slope curvature is important.

Stratification and topography data from Kaena ridge courtesy of Jody Klymak and HOME researchers (Legg, 2006)

Transient hydraulics

1 wg

FrC

U

zgC

We would expect an internal wave to be unable to propagate against the flow if

Group velocity is proportional to vertical wavelength

So we might expect the flow to be supercritical to internal tides of wavelength z < c. For Hawaiian ridge parameters, c = 465m at U0=5cm/s, so we expect transient hydraulic control of features below this scale.

To have a hydraulic jump, flow must transition from supercritical to subcritical as it flows downslope, i.e. depth change within tidal period must be significant.

Depth change is significant if 1Nh

Udx

dhUh

where

i.e. 1dx

dhN

So transient hydraulic jumps may be possible if slope is sufficiently steep

Hawaiian ridge: Dependence of overturning on slope

Real topography dh/dx(max) = 0.2, smooth dh/dx(max) = 0.2

dh/dx(max) = 0.1 dh/dx(max) = 0.06

Borelike features are found only for dh/dx(max) >> s, combined with a region of dh/dx = s at the top of the slope (Legg, 2006)

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 U0=5cm/s

U0=10cm/sSnapshots of U (color) and buoyancy (contours) just after flow reversal, for dh/dx(max) = 0.2.

Larger amplitude flow increases extent and vigor of overturning.

Influence of internal bores on dissipation

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

Steep slopes and large amplitude flows have largest dissipation.

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

Time-averaged dissipation for all simulations at location of maximum dissipation (h=-1170m)

Possible explanation of ``flow-reversal’’ mixing events (Aucan et al,2006) observed

at mooring on Hawaiian ridge flank

Potential temp

dissipation

currents

Flow-reversal mixing event

Downslope flow mixing event

Summary of progress on internal tide generation and local mixing

• Recent theoretical advances in predicting energy conversion are supported by numerical simulations.

• Only 10% of this energy is dissipated locally for most topographies

• For very narrow topography dissipation is greatly enhanced and occurs mostly in internal tide beams.

• Transient hydraulic jumps can produce a local enhancement of dissipation and mixing, when steep slopes are combined with large amplitude flows, especially when combined with breaking at critical slopes.

Most of the baroclinic energy is in the form of radiating internal tides: Q: What is their fate?

Fate of internal tides: 1. Wave-wave interactions: (a) Parametric Subharmonic Instability

At latitudes where 2f < (M2), PSI transfers energy into subharmonic with larger wavenumbers. When 2f = (M2) (at 28.9 degrees) dissipation is greatly enhanced.

MacKinnon and Winters, 2006

Fate of internal tides: 1. Wave-wave interactions: (b) steady-state continuum

Garret-Munk-like spectrum is steady state result of wave-wave interactions

(Caillol and Zeitlin, DAO, 2000)

E(,k) ~ -2m-2 for >> f

Site D spectra(Garrett and Munk)

(taken from Lvov et al, 2005)

Fate of internal tides: 2. Reflection from critical slopes: generation of internal bores

Numerical simulations demonstrate mixing is possible at all shapes of critical slopes, provided

3km

200m

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

Buoyancy field for 1st mode internal tide

1

1tan 2/1

2/1

1

wI

wI

F

Fs

1

1tan 2/1

2/1

2

wI

wI

F

Fs

21

so that reflected wave Fr > 1. Legg and Adcroft, 2003.

Fate of internal tides: 3. Scattering from corrugated slope

With corrugations, high mode structure seen in velocity profiles

Source: scattering of internal tide generated at shelf-break

Without corrugations With corrugations

Cross-slope velocities at t=3.14 M2 periods

Cross-slope velocity profiles at x=60km

Scenario: internal tide generated at shelf break, with tidal forcing U=10cm/s reflecting from continental slope: possible description of TWIST region (Nash et al, 2004)

(Legg, 2004)

Tidal energy flow chart revisited….

Wave-wave interactions

Wave-topographyinteractions

Barotropic tides

Rough topography

Internal tidesLocal 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 near 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.

PSI

Parameterizing tidal mixing in ocean models

20

)(),(

N

zFyxEq

2202

1),( ukhNyxE b

)1()(

/

/)(

H

zH

e

ezF

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 (this could be refined if DNS suggests it is necessary)

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, PSI at critical latitude.

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

Example of current tidal mixing parameterization

Parameterized diffusivity due to tidal mixing in S. Atlantic(St Laurent et al, 2002)

Conclusions

Energetically consistent parameterizations of tidal mixing are now becoming a possibility. These parameterizations must include estimates of:

1. Energy conversion from the barotropic tide;2. The fraction of this energy used for local mixing and

the spatial distribution of this mixing;3. The internal wave field generated by the tides;4. The locations of internal wave breaking, and the

mixing thus generated.

Many questions still remain, and theoretical, observational and numerical work are all needed to answer them. But we mustn’t be afraid to use what we already know, however approximate, to improve climate models!