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. Cooling. Pole. Eq. convection. overflows. Overturning. Upwelling. tidal mixing.
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Princeton University, NOAA-GFDL
Diapycnal 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.
Tidal Energy Budget in the deep ocean
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.
Tidal energy flow chart.
Global climate models do not simulate any part of this chain of events, not just the final mixing.
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)
Topography: height h, width L, depth H
Flow: speed U, oscillation frequencyw
Others: coriolis f, buoyancy frequency N
Analytical studies assume some or all of topographic/flow parameters are small – numerical simulations don’t have this restriction.
The relevant question for mixing parameterization purposes: how much energy is extracted from the barotropic tide?
St Laurent et al, 2003
Q: What happens when RL> 1, and Fr = U/(Nh) < 1?
Low, wide, shallow topo
Low, narrow, steep topo
Tall, steep topo
Baroclinic velocity snapshots from simulations of tidal flow over Gaussian topo with forcing amplitude U0=2cm/s (Legg and Huijts, 2006; using MITgcm).
Steep topography leads to generation of internal tide beams: energy concentrated on wave characteristics.
Rate of energy conversion from barotropic tide
Ratio of dissipation rate to conversion rate
St Laurent et al prediction for steep topo, h/H=0.5
St Laurent et al prediction for steep topo, deep fluid
Low, wide topography; low, narrow topography; tall wide topography; tall narrow topography
Theoretical predictions of energy conversion agree well with numerical model results.
For wider topo, only 10% of energy extracted from tide is dissipated locally; for narrow topo, much greater fraction.
All from Legg and Huijts, 2006; using MITgcm
Low wide shallow topo
Low narrow steep topo
Narrowest topo is the only case without energy peak at lowest vertical mode.
Tall steep topo
Tall steep narrow topo
(Legg and Huijts, 2006)
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.
Low narrow steep topo
Tall steep topo
Dissipation is all in narrow beams, no hydraulic effects
Possible transient internal hydraulic jumps are a location for overturning
Isopycnal deflection by large amplitude tides: U0 = 24cm/s
Large amplitude forcing over large amplitude steep topo leads to local overturns in internal hydraulic jump-like features. (Legg and Huijts, 2006)
Q: Can internal hydraulic jumps be important at more moderate (i.e. realistic) forcing velocities?
Buoyancy field for U0=5cm/s, M2 tidal forcing.
Hawaiian ridge is tall and steep.
Hydraulic jumps develop over steep slope during downslope flow: at flow reversal, jumps propagate upslope as internal bores.
Asymmetric response: slope curvature is important.
Stratification and topography data from Kaena ridge courtesy of Jody Klymak and HOME researchers (Legg, 2006)
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 lz< lc. For Hawaiian ridge parameters, lc = 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
So transient hydraulic jumps may be possible if slope is sufficiently steep
dh/dx(max) = 0.2, smooth
dh/dx(max) = 0.2
dh/dx(max) = 0.1
dh/dx(max) = 0.06
Snapshots of U(color) and buoyancy (contours) just after flow reversal, all with U0=5cm/s
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, for dh/dx(max) = 0.2.
Larger amplitude flow increases extent and vigor of overturning.
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)
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.
Flow-reversal mixing event
Downslope flow mixing event
Most of the baroclinic energy is in the form of radiating internal tides:
Q: What is their fate?
MacKinnon and Winters, 2006
At latitudes where 2f < w(M2), PSI transfers energy into subharmonic with larger wavenumbers. When 2f = w(M2) (at 28.9 degrees) dissipation is greatly enhanced.
Garret-Munk-like spectrum is steady state result of wave-wave interactions
(Caillol and Zeitlin, DAO, 2000)
E(w,k) ~ w-2m-2 for w >> f
Site D spectra
(Garrett and Munk)
(taken from Lvov et al, 2005)
Wave breaking is induced by reflection from near-critical slope, i.e. when .
Buoyancy field for 1st mode internal tide
Numerical simulations demonstrate mixing is possible at all shapes of critical slopes, provided
Legg and Adcroft, 2003.
so that reflected wave Fr > 1.
With corrugations, high mode structure seen in velocity profiles
Source: scattering of internal tide generated at shelf-break
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)
Cross-slope velocities at t=3.14 M2 periods
Cross-slope velocity profiles at x=60km
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.
Simulations have shown where internal tides will break and cause mixing – on slopes near critical angle.
For slopes within a range of critical angle
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.
G = mixing efficiency – set to 0.2 (this could be refined if DNS suggests it is necessary)
K0= 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.
E(x,y) = energy extracted from barotropic tide:
F(z) = vertical structure function:
z = vertical decay scale – set to 500m.
Does not account for preferred locations of mixing: e.g. beams, critical slopes.
Parameterized diffusivity due to tidal mixing in S. Atlantic
(St Laurent et al, 2002)
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!