The Ocean's Role In Climate

1
GFDL Climate Model Improvements as a Result of
the Entraining Gravity Current CPTStephen
Griffies, Robert Hallberg, Laura Jackson, Sonya
Legg
2
GFDL Efforts in 2004

• MOM4 Nesting high resolution submodels near
  outflows. (Griffies, Adcroft)
• HIM Improvements to channel-exchange and mixing
  parameterizations.
• Bottom Stress Driven Boundary Layer Mixing.
• Self-consistent iterations between Ri and
  mixing.
• Partially Open Faces.
• Sensitivity Studies (Jackson).
• HIM/MITgcm Idealized Intercomparisons (Legg et
  al.)

3
Requirements for adequately representing gravity
currents

• Source water must be supplied to the plume with
  the right rate and properties.
• A challenge for climate-resolution models!
• Model must be able to represent downslope flow
  without excessive numerical mixing.
• The immediate challenge for Z-coordinate models.
• Nesting and plumbing are the two approaches being
  tried for the CPT.
• Parameterizations drive mixing to the right
  extent
• Resolved energy sources of mixing
• Resolved internal shears ? Kelvin-Helmholtz
  billows, etc.
• Bottom drag on resolved flow ? 3-D turbulence
• Unresolved (omitted) energy sources of mixing
• Small-scale mean flows
• Tidal flows
• Breaking internal waves
• Subgridscale circulations need to be
  parameterized
• Baroclinic instability
• Drainage in small channels might prove important

Included but need calibration.
All of these are still missing.
All of these are still missing.
4
Source water supply

• Source Water properties depend on the right
  large-scale circulation and properties.
• Several important source waters enter through
  very narrow channels!
• Gibraltar is 12 km wide.
• Red Sea outflow channel is 5 km wide.
• Faroe Bank channel is 15km wide at depths that
  matter.

• Dardanelles 5 km wide.
• Bosporus 1 km wide.

• Channels that are much smaller than the model
  grid require special treatment.
• GFDLs Z-coordinate, B-grid MOM4 uses a diffusive
  parameterization.
• Partially open cell-faces seem to work in GFDLs
  C-grid, isopycnal coordinate HIM e.g. Gibraltar
  specified as 12 km wide in 100 km resolution
  model.
• Energy conservation dictates where to add
  metric terms.
• This problem needs work!

5
Source water supply

• Source Water properties depend on the right
  large-scale circulation and properties.
• Several important source waters enter through
  very narrow channels!
• Gibraltar is 12 km wide.
• Red Sea outflow channel is 5 km wide.
• Faroe Bank channel is 15km wide at depths that
  matter.

• Dardanelles 5 km wide.
• Bosporus 1 km wide.

• Channels that are much smaller than the model
  grid require special treatment.
• GFDLs Z-coordinate, B-grid MOM4 uses a diffusive
  parameterization.
• Partially open cell-faces seem to work in GFDLs
  C-grid, isopycnal coordinate HIM e.g. Gibraltar
  specified as 12 km wide in 100 km resolution
  model.
• Energy conservation dictates where to add
  metric terms.
• This problem needs work!

6

• A perfect model would be white everywhere i.e.
  minimal drift away from climatology.

7
Resolution requirements for avoiding numerical
entrainment in gravity currents.

• Z-coordinate
• Require that
• AND
• to avoid numerical entrainment.
• (Winton, Hallberg, and Gnanadesikan, JPO 1998)
• Suggested solutions for Z-coordinate models
• "Plumbing" parameterization of downslope flow
• Beckman Doscher (JPO 1997), Campin Goose
  (Tellus 1999).
• Adding a separate, resolved, sigma-coordinate
  boundary layer
• Gnanadesikan (1998), Killworth Edwards (JPO
  1999), Song Chao (JAOT 2000).
• Add a nested high-resolution model in key
  locations?
• No existing scheme is entirely satisfactory!
• Sigma-coordinate Avoiding entrainment requires
  that
• Isopycnal-coordinate Numerical entrainment is
  not an issue - BUT
• If resolution is inadequate, no entrainment can
  occur. Need

8
Gravity Current Diapycnal Mixing in Isopycnic
Coordinates

• Mixing driven by unresolved flows (internal
  waves, etc.) are parameterized with a background
  diffusivity.
• Negligible for gravity currents!
• Entrainment in the Interfacial Layer at the top
  of the plume is driven by energy extracted from
  the mean shear, as in Kelvin-Helmholtz billows
  and similar processes.
• Described with resolved shear Richardson-number
  dependent parameterizations.
• Currently Ellison Turner (1959) bulk
  parameterization reinterpreted for a shear Ri
• - or The Richardson number dependent part of
  KPP (Large et al., 1994)
• Rotation and vertical nonlocality are not
  included.
• Plumes have well-mixed near bottom regions.
• Energy extracted by bottom drag suggests a
  reasonable parameterization
• There is clearly room for improvement in the
  parameterization of the physical processes drive
  mixing in gravity currents!

9
Examples of shear Richardson number
parameterizations

• Generic shear parameterizations e.g. KPP (Large
  et al., 1994)
• Typically calibrated for the Equatorial
  Undercurrent.
• Plume-specific parameterizations e.g. Ellison
  Turner (1959) bulk Ri parameterization
  reinterpreted for shear Ri
• This can be cast as a diffusivity, D is over an
  unstable region

Need Calibration Against Overflows! May Need
Resolution Dependence!
10
Net Gravity Current Entrainment for 6 Test Cases
in Z- and Isopycnal-Coordinate Models (Legg et
al, Ocean Modelling, in press)

• Nonhydrostatic Z-coordinate model (MITgcm -
  solid) spurious mixing decreases, then resolved
  mixing increases with increasing resolution.
• Isopycnal-coordinate model (HIM - dashed)
  parameterized mixing increases with increasing
  resolution.

Final Plume Buoyancy
Horizontal Resolution
11
Observed profiles from Red Sea plume from
RedSOXcourtesy H. Peters.
Actively mixing Interfacial Layer Shear Ri
Param. Appropriate Here.
Well-mixed Bottom Boundary Layer
12
Bottom Boundary Layer Mixing

• Diapycnal mixing of density requires work.
• The rate at which bottom drag extracts energy
  from the resolved flow is straightforward to
  calculate.
• Assumptions
• 20? of the extracted energy is available to
  drive mixing.
• Available work decays away from the bottom with
  e-folding scale of
• Mixing completely homogenizes the near bottom
  water until the energy source is exhausted.
• Legg, Hallberg, Girton, Ocean Modelling, 2004?

13
500 m x 30 m MITgcm
With thick plumes, both Interfacial and and
Drag-induced Mixing are needed. (Legg et al.,
Ocean Modelling, 2004?)
10 km x 25 layer HIM
Ellison Turner Drag Mixing
Ellison Turner Mixing Only
14

• Double Mediterranean Plumes

15
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16
Gravity Current Entrainment A Sensitivity Study.
17

• Which parameters affect entrainment?
• cdrag Coefficient of bottom friction ref
  value 0.002
• bbl_effic Efficiency with which bottom boundary
  layer energy drives diapycnal mixing ref
  value 0.2
• Ri_crit Critical Richardson number at which
  the current starts entraining ref value
  0.8
• E_mag Magnitude of entrainment ref value
  0.01
• The entrainment rate is determined by Gradient
  Richardson number

18

• Plume Properties
• Downstream of the entrainment we measure
  tracer-weighted averages of
• buoyancy
• depth of plume
• distance offshore
• and then average in time and distance along the
  plume path over the region where the buoyancy is
  statistically steady.
• Also calculate the corner entrainment within a
  box, defined by Legg et al (2004) as

Transport out of/into box by tracer
Root mean square velocity
Surface area of tracer
19

• Model Setup
• HIM (Hallberg Isopycnal Model)
• 10km resolution, 25 layers
• sponges to remove passive tracer and damp
  density structure to ambient state
• no slip, no flux boundary conditions
• constant f
• quadratic drag for bottom stress
• biharmonic Smagorinsky viscosity
• gradient-Richardson number parameterisation for
  entrainment (see Hallberg, 2000)
• bottom boundary layer mixing parameterisation
  (see Legg et al, 2004)
• DOME topography
• imposed inflow in geostrophic balance density
  of the plume is equal to ambient density at the
  bottom of the slope.

20
Topography of Case 1 Different cases
no ambient strat.
different buoyancies/strat.
weaker gradient
no rotation
reference case
21
Case 1 (ref)
Sensitivity to parameters
22
Case 4 (no rotation)
23
Case 4 (no rotation)
Case 4 with bbl_effic0
24
Case 5 (weaker slope)
25
Cross sections
Case 1
Case 5
26
Case 6 (no ambient strat.)
27
Sensitivity to Ri_crit
28
Sensitivity to cdrag
29
Comparison with z-coordinate models from Legg et
al (2004)
Final buoyancy
Resolution
30
Conclusions

• The final density is predominantly controlled by
  the critical Richardson number
• The bottom friction controls the location of the
  plume (when rotation is included) which could
  have an indirect effect on the density with
  realistic topography.
• The magnitude (rate) of the entrainment has
  little effect
• The efficiency of the bottom boundary layer
  mixing has little effect in the cases with
  rotation and, in the case without rotation, if it
  is sufficiently large. When the mixing is
  switched off in case 4 there is a large increase
  in the final density and depth due to the
  formation of two plumes.
• With no ambient stratification, increasing the
  critical Richardson number creates a less dense,
  but deeper plume.

31
Case 2
32
Case 3
33
Sensitivity to Ent_mag
34
Sensitivity to bbl_effic
35
Sensitivity to bbl_effic bbl_effic 0.0, 0.1,
0.2, 0.3 Tracer layer avg
std zero
low ref high zero
low ref high 1 15.799 15.773
15.824 15.787 2.315 2.352 2.324
2.342 2 16.355 16.355 16.355 16.355
2.292 2.292 2.292 2.292 3
16.036 16.036 16.036 16.036 2.860
2.860 2.860 2.860 4 13.798 12.591
12.840 12.601 5.057 4.053
3.515 2.818 5 16.789 16.789 16.789
16.867 2.488 2.488 2.488
2.541 6 6.960 7.049 6.777 7.200
2.037 2.071 2.048 2.095 Tracer
depth avg
std zero low
ref high zero low ref
high 1 1570.76 1570.48 1584.29 1572.82
522.48 536.58 538.84 533.53 2 1563.85
1563.85 1563.85 1563.85 541.64
541.64 541.64 541.64 3 1087.09 1087.09
1087.09 1087.09 329.93 329.93 329.93
329.93 4 1863.18 1676.92 1717.13
1677.48 793.71 638.64 554.10
401.57 5 1445.70 1445.70 1445.70 1413.54
579.02 579.02 579.02 572.84 6
2148.74 2130.45 2184.69 2099.53 943.68
940.49 942.41 941.41 Tracer distance
avg
std zero low ref
high zero low ref high 1
113.38 113.48 115.57 114.23 52.21
53.43 54.23 53.42 2 109.36
109.36 109.36 109.36 54.39 54.39
54.39 54.39 3 55.51 55.51 55.51
55.51 32.00 32.00 32.00
32.00 4 1056.02 1057.86 1033.78 1056.09
164.27 231.84 204.77 282.56 5 188.48
188.48 188.48 182.02 113.60
113.60 113.60 111.77 6 176.96 174.69
180.23 171.9 100.71 100.23 101.06
101.98 Corner entrainment zero
low ref high 1 0.001038
0.001033 0.001045 0.001040 2 0.000809
0.000809 0.000809 0.000809 3 0.000361
0.000361 0.000361 0.000361 4 0.004207
0.004400 0.004478 0.004372 5 0.000666
0.000666 0.000666 0.000602 6 0.001572
0.001679 0.001645 0.001622
36
Sensitivity to friction CDRAG 0.0006, 0.002,
0.006 Tracer layer avg
std low ref
high low ref high 1
15.341 15.824 15.907 2.461 2.324
2.216 2 15.848 16.355 16.043
2.500 2.292 2.172 3 15.771 16.036
16.054 2.783 2.860 2.802 4
12.458 12.840 13.216 3.198 3.515
3.987 5 16.231 16.789 16.606
2.598 2.488 2.338 6 6.959 6.777
6.966 2.238 2.048 1.791 Tracer
depth avg
std low ref high
low ref high 1 1363.94 1584.29
1761.51 525.44 538.84 503.80 2
1397.55 1563.85 1709.90 509.65
541.64 483.22 3 1015.12 1087.09 1269.65
324.60 329.93 319.55 4 1658.77
1717.13 1739.64 509.89 554.10
672.63 5 1264.51 1445.70 1578.40
525.14 579.02 551.68 6 1940.48 2184.69
2538.79 956.77 942.41 928.15
Tracer distance avg
std low ref
high low ref high 1 89.45
115.57 145.15 50.63 54.23
58.08 2 91.63 109.36 135.28
50.27 54.39 53.82 3 48.37 55.51
78.74 31.30 32.00 33.59 4
1027.54 1033.78 1047.03 207.72
204.77 248.45 5 151.18 188.48 223.75
100.80 113.60 109.53 6 151.47
180.23 252.51 101.20 101.06
141.46 Corner entrainment low ref
high 1 0.001043 0.001045 0.001120
2 0.001157 0.000809 0.0009 3
0.000841 0.000361 0.000323 4 0.004317
0.004478 0.004501 5 0.000535 0.000666
0.000556 6 0.00169 0.001645 0.001778

37
Sensitivity to ent_const ent_const 0.005,
0.001, 0.002 Tracer layer avg
std low
ref high low ref high 1
15.929 15.824 15.735 2.404
2.324 2.322 2 16.475 16.355 16.187
2.374 2.292 2.279 3 16.321
16.036 15.891 2.895 2.860
2.848 4 12.659 12.840 12.451
3.441 3.515 3.518 5 16.934 16.789
16.586 2.535 2.488 2.432 6
7.033 6.777 6.827 2.030 2.048
1.950 Tracer depth avg
std low ref
high low ref high 1
1584.06 1584.29 1555.86 533.89
538.84 517.78 2 1585.74 1563.85 1519.59
542.18 541.64 534.61 3 1108.59
1087.09 1084.10 349.34 329.93
352.36 4 1688.07 1717.13 1617.75
538.54 554.10 590.42 5 1408.22 1445.70
1423.84 551.04 579.02 561.38 6
2156.44 2184.69 2206.13 951.49
942.41 993.56 Tracer distance
avg std low
ref high low ref
high 1 115.05 115.57 112.33 52.71
54.23 51.70 2 111.45 109.36
104.72 54.50 54.39 53.41 3
57.51 55.51 55.20 33.96 32.00
34.14 4 1039.64 1033.78 1039.43
186.23 204.77 308.06 5 180.11 188.48
185.06 107.11 113.60 109.74 6
179.55 180.23 187.87 104.32 101.06
113.53 Corner entrainment low
ref high 1 0.000988 0.001045
0.001055 2 0.000839 0.000809 0.000776
3 0.000381 0.000361 0.000446 4
0.004257 0.004478 0.004504 5 0.000435
0.000666 0.000561 6 0.001632 0.001645
0.001693
38
Sensitivity to Rino_crit Rino_crit 0.6, 0.8,
1.0 Tracer layer avg
std low ref
high low ref high 1 16.545
15.824 15.277 2.442 2.324
2.225 2 16.905 16.355 15.587
2.447 2.292 2.275 3 16.504 16.036
15.660 3.010 2.860 2.818 4
13.291 12.840 12.173 3.304 3.515
3.346 5 17.600 16.789 16.154
2.551 2.488 2.458 6 7.658 6.777
6.396 2.209 2.048 1.822 Tracer
depth avg
std low ref high
low ref high 1 1672.82 1584.29
1513.75 544.21 538.84 508.68 2
1610.99 1563.85 1449.28 551.00
541.64 493.15 3 1165.93 1087.09 1000.35
374.67 329.93 296.87 4 1784.37
1717.13 1569.72 513.67 554.10
585.21 5 1494.61 1445.70 1354.63
599.82 579.02 537.76 6 2190.46 2184.69
2266.80 932.61 942.41 953.89
Tracer distance avg
std low ref
high low ref high 1 122.32
115.57 109.59 54.19 54.23
50.85 2 112.61 109.36 98.84
54.59 54.39 49.09 3 62.78 55.51
47.80 36.51 32.00 28.43 4
1028.26 1033.78 1027.39 220.35
204.77 275.87 5 196.36 188.48 172.86
117.71 113.60 103.21 6 181.32
180.23 194.57 104.47 101.06
107.32 Corner entrainment low ref
high 1 0.000826 0.001045 0.001215
2 0.000729 0.000809 0.001069 3
0.000396 0.000361 0.000399 4 0.003429
0.004478 0.003148 5 0.000409 0.000666
0.000667 6 0.001272 0.001645 0.002123
39

• Several important marginal sea connections are
  constrained by very narrow channels!
• Gibraltar 12 km wide.
• Red Sea outflow channel 5 km wide.
• Dardanelles 5 km wide.
• Bosporus 1 km wide.
• Representing these with the grid-resolution can
  lead to wildly incorrect exchange of water!

40
Partially open cell-faces

• Partially open cell-faces seem to work nicely in
  HIM e.g. Gibraltar specified as 12 km wide in
  100 km resolution model. (Suggested by Alistair
  Adcroft.)
• Energy conservation dictates where metric terms
  are altered to put in this information.

41
Modifications to Shallow Water Equations

• E.g. Sadournys energy conserving discretization
  of the inviscid shallow water equations.
• Terms underlined in red are affected directly by
  using the partially open faces.
• Terms underlined in green are affected
  indirectly (i.e. no code changes).
• The derivation of this closely follows the
  appendix of Arakawa Lamb (1982).

42
GFDL Efforts in 2004

• MOM4 Nesting high resolution submodels near
  outflows.
• Substantial progress has been made on the ability
  to run common global experiments.
• Work on nesting is well underway, but no
  simulations have been completed.
• S. Griffies will be on sabbatical in 2005, but
  Adcroft (now at GFDL) will be working actively on
  Z-coordinate representations of overflows.
• Need thorough assessment of how available options
  perform in a realistic climate model with (for
  example) Gent-McWilliams parameterizatons.
• HIM Improvements to channel-exchange and mixing
  parameterizations.
• Bottom Boundary Layer Mixing.
• Energy extracted from resolved flow by drag
  drives mixing against the bottommost
  stratification.
• Particularly important with nonrotating flows,
  but some global simulations show an impact.
• Self-consistent iterations between Ri and
  mixing.
• Shears gt Ri gt Entrainment gt Shears
• Iterations now converge to consistent solutions.
• Partially Open Faces.
• Critical for representing Strait exchanges.
• Sensitivity Studies (Jackson).
• HIM/MITgcm Idealized Intercomparisons (Legg et
  al.)