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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!