Vertical Mixing Parameterizations and their effects on the skill of Baroclinic Tidal Modeling Robin...
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Vertical Mixing Parameterizations and their effects on the skill of Baroclinic Tidal Modeling Robin Robertson Lamont-Doherty Earth Observatory of Columbia University Palisades, NY
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Transcript of Vertical Mixing Parameterizations and their effects on the skill of Baroclinic Tidal Modeling Robin...
Vertical Mixing Parameterizations
and their effects on the skill of
Baroclinic Tidal Modeling
Robin Robertson
Lamont-Doherty Earth Observatory
of Columbia University
Palisades, NY
Domain
-128.2 -128.0 -127.8 -127.6 -127.4
Longitude
32.0
32.2
32.4
32.6
32.8
Lat
itu
de
500
1000
1500
2000
2500
3000
3500
4000
W ater Depth
(m)
x = 2 km y =2 km 60 levels
taken from Smith and Sandwell [1997]
B2
B3
F2
F3F4,F5 C
R3
R2 P
Internal Wave Theory
• Internal wave generation criteria according to linear theory
- slope of internal wave rays =1 – critical
– Most generation – resonant
< 1 – subcritical– Less generation– Propagates both on and offslope
> 1 – supercritical– Less generation– Propagates offslope
H
y
122 2
2 2
f
N
Internal Tide Generation according to linear theory
-128.2 -128.0 -127.8 -127.6 -127.4 -127.2
Longitude
32.0
32.2
32.4
32.6
32.8
33.0
La
titu
de
M2
-128.2 -128.0 -127.8 -127.6 -127.4 -127.2
Longitude
32.0
32.2
32.4
32.6
32.8
33.0S
2
-128.2 -128.0 -127.8 -127.6 -127.4 -127.2
Longitude
27.0
27.2
27.4
27.6
27.8
28.0
Lat
itu
de
-128.2 -128.0 -127.8 -127.6 -127.4 -127.2
Longitude
27.0
27.2
27.4
27.6
27.8
28.0K
1O
1
Red indicates expected internal tide generation
M2 Baroclinic Tides
-4000
-2000
0
Dep
th (
m)
0.5
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Bottom
-4000
-2000
0
Dep
th (
m)
(cm s -1 )
-4000
-2000
0
Dep
th (
m)
-4000
-2000
0
Dep
th (
m)
32.0 32.2 32.4 32.6 32.8Latitude
-4000
-2000
0
Dep
th (
m)
128o 1.7' W
127o 51.4'W
127o 41.0' W
127o 30.7'W
127o 20.4'W
Base Case (1)
K1 Baroclinic Tides
-4000
-2000
0
De
pth
(m
)
0.5
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Bottom
-4000
-2000
0
De
pth
(m
)
(cm s -1 )
-4000
-2000
0
De
pth
(m
)
-4000
-2000
0
De
pth
(m
)
32.0 32.2 32.4 32.6 32.8Latitude
-4000
-2000
0
De
pth
(m
)
128o 1.7' W
127o 51.4'W
127o 41.0' W
127o 30.7'W
127o 20.4'W
Base Case (1)
Comparison to Observations:
M2 Major Axes Observations Model Results
32 42.05 N128 2.70 W
32 28.64 N127 50.44 W
32 28.27 N127 48.57 W
32 26.56 N127 46.16 W
32 9.30 N128 0.49 W
32 25.75N127 50 W
32 24.61 N127 47.57 W
Major Axis (cm s -1 )
M2
B2 F2 R2 C
B3 F3 R3
0 4 8
0
500
1000
1500
2000
2500
3000
3500
4000
4500
De
pth
(m
)
0 4 8 120 4 8 0 4 8 12
0 4 8
0
500
1000
1500
2000
2500
3000
3500
4000
4500
De
pth
(m
)
0 4 8 0 4 8 120 4 8 12
P 32 26.56 N127 46.70 W
Comparison to Observations:K1 Major Axes
Observations Model Results
32 42.05 N128 2.70 W
32 28.64 N127 50.44 W
32 28.27 N127 48.57 W
32 26.56 N127 46.16 W
32 9.30 N128 0.49 W
32 25.75N127 50 W
32 24.61 N127 47.57 W
Major Axis (cm s -1 )
K1
B2 F2 R2 C
B3 F3 R3
0 4 8 12
0
500
1000
1500
2000
2500
3000
3500
4000
4500
De
pth
(m
)
0 4 8 12 160 4 0 4 8 12 16 20
0 4
0
500
1000
1500
2000
2500
3000
3500
4000
4500
De
pth
(m
)
0 4 8 12 0 4 8 12 160 4 8 12
P 32 26.56 N127 46.70 W
Comparison to Observations:Mean Currents
Observations Model Results
32 42.05 N128 2.70 W
32 28.64 N127 50.44 W
32 28.27 N127 48.57 W
32 26.56 N127 46.16 W
32 9.30 N128 0.49 W
32 25.75N127 50 W
32 24.61 N127 47.57 W
Major Axis (cm s -1 )
B2 F2 R2 C
B3 F3,F4,F5
R3
0 4 8 12 16
0
500
1000
1500
2000
2500
3000
3500
4000
4500
De
pth
(m
)
0 4 8 12 16 200 4 0 4 8 12 16 20
0 4
0
500
1000
1500
2000
2500
3000
3500
4000
4500
De
pth
(m
)
0 4 8 120 4 8 12 16 0 4 8 12 16
P 32 26.56 N127 46.70 W
Vertical Mixing Parameterizations
• Large-McWilliams-Doney (LMD) Kp profile
• Mellor-Yamada 2.5 level turbulence closure (MY2.5)
• Brunt-Väisälä frequency (BVF)• Pacanowski-Philander (PP)• Generic Length Scale (GLS)
• Lamont Ocean Atmosphere Mixed Layer Model (LOAM )
• LMD - modified
Large-McWilliams-Doney Kp profile
• Primary processes– Local Ri instabilities due to resolvable
vertical shear• If (1-Ri/0.7) > 0
10-3 (1-Ri/0.7)3
– Convection• N dependent 0.1 * [1.-(2x10-5 –N2)/2x10-5]
– Internal wave • N dependent 10-6/N2 (min N of 10-7)
– Double diffusion• Only for tracers
• For Ri < 0.8, the first dominates• For Ri > 0.8, the third dominates• Modified
– Non-local fluxes, Langmuir, Stokes drift
– Changes two of the Kp profile values
Mellor-Yamada 2.5 level turbulence closure
• Designed for boundary layer flows
• Based on turbulent kinetic energy and length scale which are time stepped through the simulation
• Matched laboratory turbulence• Logarithmic law of the wall• Not designed for internal wave
mixing• Fails in the presence of
stratification
Brunt-Väisälä frequency
• Diffusivity is a function of N– If N < 0 Kv = 1– If N = 0 Kv = background value– If N > 0 Kv = 10-7/N
• Min of 3x10-5
• Max of 4x10-4
– Background values is input (10-6)
Pacanowski-Philander
• Designed for the tropics• Gradient Ri dependent
– If Ri > .2 Kv = 0.01/(1-5Ri)2+background
max = 0.01– Otherwise Kv = 0.01
• LOAM – version modified for use outside the tropics– If Ri > .2 Kv = 0.05/(1-5Ri)2+background
max = 0.05– Otherwise Kv = background
Generic Length Scale
• Two generic equations
– D - turbulent and viscous transport
– P - KE production by shear
– G - KE production by buoyancy - Dissipation
– c - model constants
• Based on turbulent kinetic energy and length scale which are time stepped through the simulation
• MY2.5 is a special case – p=0, m=1, n=1
cGcPc
kDut xi
i 231
nmp
kc0
Major Axis Errors
Vertical Mixing Parameterization
M2 (cm s-1)
S2 (cm s-1)
O1 (cm s-1)
K1 (cm s-1)
LMD Mixing 1.80 0.70 1.04 4.06 Brünt-Väisäla
Frequency Mixing 1.92 0.78 1.22 4.05
Mellor-Yamada 2.5 Level Closure
1.86 0.85 1.13 3.97
Pacanowski-Philander Mixing
1.94 0.64 1.37 4.33
GLS Mixing 1.79 0.76 1.20 4.11 LOAM Mixing 1.92 0.63 1.31 4.37
LMD without BKPP 1.86 0.78 1.21 3.92 Modified LMD 1.77 0.80 1.09 4.02
Red indicated absolute error values lower than those of the base case.
Comparisons to Observations (velocities)
Observations Model Results
32 25.75 N127 50.0 W
32 25.75 N127 50.0 W
32 26.56 N127 46.70 W
Major Axis (cm s -1 )
M2
LMD MY2.5 BVF PP
0 4 8
0
500
1000
1500
De
pth
(m
)
0 4 8 0 4 80 4 8 0 4 832 26.56 N127 46.70 W
0 4 8 12
0
200
400
600
De
pth
(m
)
0 4 8 120 4 8 12 0 4 8 12 0 4 8 12
K1
GLS
0 4 8 12
1500
1000
500
0
De
pth
(m
)
0 4 8 12 0 4 8 12 0 4 8 12 0 4 8 12
0 8 16
600
400
200
0
De
pth
(m
)
0 8 16 0 8 16 0 8 16 0 8 16
Flank (F3,F4,F5)
Center (P)
Flank (F3,F4,F5)
Center (P)
Vertical Diffusivity Observations
From Kunze et al. [1991]
Vertical Diffusivity (Temperature)
-4000
-2000
0
Dep
th (
m)
1E-006
1E-005
5E-005
1E-004
5E-004
1E-003
5E-003
Bottom
-4000
-2000
0
De
pth
(m
)
(m 2 s -1 )
-4000
-2000
0
Dep
th (
m)
-4000
-2000
0
Dep
th (
m)
32.0 32.2 32.4 32.6 32.8Latitude
-4000
-2000
0
Dep
th (
m)
127o 41.0' W
127o 41.0'W
127o 41.0' W
127o 41.0'W
127o 41.0'W
LMD
PP
BVF
MY2.5
GLS
Vertical Diffusivity (Temperature) (cont)
-4000
-2000
0
Dep
th (
m)
1E-006
1E-005
5E-005
1E-004
5E-004
1E-003
5E-003
Bottom
-4000
-2000
0
Dep
th (
m)
(m 2 s -1 )
-4000
-2000
0
Dep
th (
m)
-4000
-2000
0
Dep
th (
m)
32.0 32.2 32.4 32.6 32.8Latitude
-4000
-2000
0
Dep
th (
m)
127o 41.0' W
127o 41.0'W
127o 41.0' W
127o 41.0'W
127o 41.0'W
LOAM
PP
LMDno BKPP
LMDmodified
LMD
Vertical Diffusivity Observations
Vertical Diffuxivity (Temperature)
-1000
-500
0
Dep
th (
m)
1E-006
1E-005
5E-005
1E-004
5E-004
1E-003
5E-003
Bottom
-1000
-500
0
De
pth
(m
)
(m 2 s -1 )
-1000
-500
0
Dep
th (
m)
-1000
-500
0
Dep
th (
m)
32.40 32.42 32.44 32.46 32.48 32.50 32.52 32.54Latitude
-1000
-500
0
Dep
th (
m)
127o 41.0' W
127o 41.0'W
127o 41.0' W
127o 41.0'W
127o 41.0'W
LMD
PP
BVF
MY2.5
GLS
Vertical Diffusivity (Temperature)
-4000
-2000
0
Dep
th (
m)
1E-006
1E-005
5E-005
1E-004
5E-004
1E-003
5E-003
Bottom
-4000
-2000
0
De
pth
(m
)
(m 2 s -1 )
-4000
-2000
0
Dep
th (
m)
-4000
-2000
0
Dep
th (
m)
32.0 32.2 32.4 32.6 32.8Latitude
-4000
-2000
0
Dep
th (
m)
127o 41.0' W
127o 41.0'W
127o 41.0' W
127o 41.0'W
127o 41.0'W
LMD
PP
BVF
MY2.5
GLS
Summary
• Baroclinic tides were simulated using ROMS
• Semidiurnal tides were reproduced successfully
• Diurnal tides were not reproduced – Critical latitude effects– Mean currents insufficiently
simulated
• Generic Length Scale (GLS) produced the most realistic vertical diffusivities
• Acknowledgments – Data from Brink, Toole, Kunze, Noble, and Eriksen
Model Description Regional Ocean Modeling System (ROMS) Primitive equation model; non-linear Split 2-D and 3-D modes Boussinesq and hydrostatic approximations Horizontal advection - 3rd order upstream
differencing [McWilliams and Shchepetkin] Explicit vertical advection Laplacian lateral diffusion along sigma
surfaces (1 m2 s-1) LMD scheme for vertical mixing Exact baroclinic pressure gradient Density based on bulk modulus Tidal Forcing – M2, S2, O1, and K1
Elevations - set at boundaries 2-D velocities – radiation [Flather] 3-D velocities – flow relaxation scheme tracers – flow relaxation scheme Time Step - 4 s barotropic, 120 s baroclinc
mode Simulation Duration: 30 days
Hydrography
32.0 32.2 32.4 32.6 32.8
Latitude
-4000.0
-3000.0
-2000.0
-1000.0
0.0
Dep
th (
m)
01234567891011121416
(o C)
from Levitus et al [1999]
Bottom
Vertical Cut at 127o 51.4' W
Evaluation of Operational Considerations and Parameterizations
Horizontal Resolution: Improving resolution improves agreement 1 km shows best agreement
Vertical Resolution: No. of Levels: Doubling the number of levels from 30 to 60 slightly
improved the agreement Increasing the number of levels to 90, showed no
improvement Spacing: Uneven spacing with more levels near the
surface and bottom improves agreement with observations
Best match - shallow mixed layer, S = 2, and B
= .5 Bathymetry:
Improvement with the finer scale Eriksen bathymetry Increased generation of internal tides on a small scale
Hydrography: Improvement with the finer scale Kunze hydrography
Baroclinic Pressure Gradient: Weighted Density Jacobian performed more poorly than
Spline Density Jacobian Vertical Mixing:
GLS showed the best agreement Horizontal Mixing: No appreciable effect
Sensitivity Study
• Bathymetry
• Hydrography
• Horizontal Resolution
• Vertical Resolution and Spacing
• Baroclinic Pressure Gradient Parameterization
• Vertical Mixing Parameterization
• Horizontal Mixing
Major Axis ErrorsParameter
Investigated
M2
(cm s-1) S2
(cm s-1) O1
(cm s-1) K1
(cm s-1)
Base Case
x, y =2 km; 60 levels;
unevenly spaced; Splines Density Jacobian BPG;
LMD Mixing
1.84
0.67
1.15
4.16
x, y =4 km (30 levels)
2.35 0.65 1.23 4.43 Horizontal Resolution x, y =1 km
(60 levels) 1.57 0.56 1.04 3.57
Bathymetry Smith & Sandwell
2.04 0.58 1.16 3.95
Hydrography Kunze 1.60 0.91 1.25 4.15 30 levels 1.89 0.76 1.39 4.15
90 levels 1.97 0.65 1.58 4.87 60 levels;
thermocline = 400m
S=1, B=1
1.91 0.66 1.36 4.23
60 levels; thermocline = 100m
S=1, B=1
1.85 0.76 1.28 4.07
60 levels; thermocline = 100m
S=2, B=1
1.88 0.67 1.35 4.17
Vertical Resolution and
Spacing
60 levels; thermocline = 100m
S=4, B=1
1.87 0.75 1.29 4.04
Baroclinic Pressure Gradient
Weighted Density Jacobian BPG
2.07 0.59 1.06 4.19
Brünt-Väisäla Frequency Mixing
1.98 0.76 1.32 4.14
Mellor-Yamada 2.5 Level Closure
1.92 0.83 1.23 4.07
Pacanowski-Philander Mixing
1.83 0.77 1.30 4.13
LOAM Mixing 1.92 0.74 1.39 4.23 LMD without BKPP 1.91 0.76 1.31 4.02
Modified LMD 1.81 0.78 1.20 4.12
Vertical Mixing
GLS Mixing 1.83 0.73 1.30 4.20
Red indicated absolute error values lower than those of the base case.
Bathymetry
-128.1 -127.9 -127.7 -127.5
Longitude
32.1
32.3
32.5
Lat
itu
de
1000
1500
2000
2500
3000
3500
4000
Smith and Sandwell fine scale from Eriksen
-128.1 -127.9 -127.7 -127.5
Longitude
32.1
32.3
32.5
32.7
Lat
itu
de
(m )
Bathymetry- M2
-4000
-2000
0
Dep
th (
m)
0.5
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Bottom
-4000
-2000
0
Dep
th (
m)
(cm s -1 )
-4000
-2000
0
Dep
th (
m)
32.00 32.20 32.40 32.60 32.80
Latitude
-4000
-2000
0
Dep
th (
m)
127o 51.4' W
127o 51.4'W
127o 41.0' W
127o 41.0'W
S&S
Eriksen
S&S
Eriksen
Bathymetry- K1
-4000
-2000
0
Dep
th (
m)
0.5
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Bottom
-4000
-2000
0
Dep
th (
m)
(cm s -1 )
-4000
-2000
0
Dep
th (
m)
32.00 32.20 32.40 32.60 32.80
Latitude
-4000
-2000
0
Dep
th (
m)
127o 51.4' W
127o 51.4'W
127o 41.0' W
127o 41.0'W
S&S
Eriksen
S&S
Eriksen
Horizontal Resolution – M2
-4000
-2000
0
Dep
th (
m)
0.5
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Bottom
-4000
-2000
0
De
pth
(m
)
(cm s -1 )
-4000
-2000
0
De
pth
(m
)
-4000
-2000
0
Dep
th (
m)
32.0 32.2 32.4 32.6 32.8 33.0Latitude
-4000
-2000
0
De
pth
(m
)
127o 51.4' W
127o 51.4'W
127o 41.0' W
127o 41.0'W
127o 41.0'W
4 km
1km
4 km
2 km
1 km
Horizontal Resolution – K1
-4000
-2000
0
Dep
th (
m)
0.5
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Bottom
-4000
-2000
0
Dep
th (
m)
(cm s -1 )
-4000
-2000
0
Dep
th (
m)
-4000
-2000
0
Dep
th (
m)
32.0 32.2 32.4 32.6 32.8 33.0Latitude
-4000
-2000
0
Dep
th (
m)
127o 51.4' W
127o 51.4'W
127o 41.0' W
127o 41.0'W
127o 41.0'W
4 km
1km
4 km
2 km
1 km
Comparison to Observations M2Observations 2 km
32 42.05 N128 2.70 W
32 28.64 N127 50.44 W
32 28.27 N127 48.57 W
32 26.56 N127 46.16 W
32 9.30 N128 0.49 W
32 25.75N127 50 W
32 24.61 N127 47.57 W
Major Axis (cm s -1 )
B2 F2 R2 C
B3 F3 R3
0 4 8
0
500
1000
1500
2000
2500
3000
3500
4000
4500
De
pth
(m
)
0 4 8 120 4 8 0 4 8 12
0 4 8
0
500
1000
1500
2000
2500
3000
3500
4000
4500
De
pth
(m
)
0 4 8 0 4 8 120 4 8 12
P 32 26.56 N127 46.70 W
1 km 4 km
Comparison to Observations K1Observations 2 km
32 42.05 N128 2.70 W
32 28.64 N127 50.44 W
32 28.27 N127 48.57 W
32 26.56 N127 46.16 W
32 9.30 N128 0.49 W
32 25.75N127 50 W
32 24.61 N127 47.57 W
Major Axis (cm s -1 )
B2 F2 R2 C
B3 F3 R3
0 4 8 12
0
500
1000
1500
2000
2500
3000
3500
4000
4500
De
pth
(m
)
0 4 8 12 160 4 0 4 8 12 16 20
0 4
0
500
1000
1500
2000
2500
3000
3500
4000
4500
De
pth
(m
)
0 4 8 12 0 4 8 12 160 4 8 12
P 32 26.56 N127 46.70 W
1 km 4 km
Comparison to Observations Mean Currents
Observations 2 km
32 42.05 N128 2.70 W
32 28.64 N127 50.44 W
32 28.27 N127 48.57 W
32 26.56 N127 46.16 W
32 9.30 N128 0.49 W
32 25.75N127 50 W
32 24.61 N127 47.57 W
Major Axis (cm s -1 )
B2 F2 R2 C
B3 F3 R3
0 10 20 30
0
500
1000
1500
2000
2500
3000
3500
4000
4500
De
pth
(m
)
0 10 200 10 0 10 20 30
0 10
0
500
1000
1500
2000
2500
3000
3500
4000
4500
De
pth
(m
)
0 4 8 12 0 10 200 10 20 30
P 32 26.56 N127 46.70 W
1 km 4 km
Baroclinic Pressure Gradient
Bottom
-4000
-2000
0
Dep
th (
m)
(cm s -1 )
-4000
-2000
0
Dep
th (
m)
-4000
-2000
0
Dep
th (
m)
32.0 32.2 32.4 32.6 32.8Latitude
-4000
-2000
0
De
pth
(m
)
127o 41.0'W
127o 41.0' W
127o 41.0'W
127o 41.0'W
0.5
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0M2
M2
K1
K1
SDJ
W DJ
SDJ
W DJ
Simulations
Case Number
Purpose
Horizontal Resolution
(x, y)
Vertical
Resolution no. of levels)
Vertical Resolution:
Spacing(mixed layer,
S B)
Baroclinic Pressure Gradient
Vertical Mixing
Horizont
al Mixing
Other
1 Base Case 2 km 60 uneven(100,2,.5)
SDJ LMD 2nd Order
Laplacian
2 Horizontal Resolution
4 km 30 uneven(100,2,.5)
SDJ LMD 2nd Order
Laplacian
3 Horizontal Resolution
1 km 60 uneven(100,2,.5)
SDJ LMD 2nd Order
Laplacian
4 Bathymetry 2 km 60 uneven(100,2,.5)
SDJ LMD 2nd Order
Laplacian
Smith &
Sandwell5 Hydrography 2 km 60 uneven
(100,2,.5)SDJ LMD 2nd
Order Laplacia
n
Kunze
6 Vertical Resolution
2 km 30 uneven(100,2,.5)
SDJ LMD 2nd Order
Laplacian
7 Vertical Resolution
2 km 90 uneven(100,2,.5)
SDJ LMD 2nd Order
Laplacian
8 Vertical Resolution
2 km 60 even (400, 1, 1)
SDJ LMD 2nd Order
Laplacian
9 Vertical Resolution
2 km 60 even (100, 1, 1)
SDJ LMD 2nd Order
Laplacian
10 Vertical Resolution
2 km 60 uneven (100, 2, 1)
SDJ LMD 2nd Order
Laplacian
11 Vertical Resolution
2 km 60 uneven (100, 4, 1)
SDJ LMD 2nd Order
Laplacian
12 Baroclinic Pressure Gradient
2 km 60 uneven(100,2,.5)
Weighted Density
Jacobian
LMD 2nd Order
Laplacian
13 Vertical Mixing
2 km 60 uneven(100,2,.5)
SDJ Brünt-Väisäla Frequency
2nd Order
Laplacian
14 Vertical Mixing
2 km 60 uneven(100,2,.5)
SDJ Mellor-Yamada 2.5 Level Clos.
2nd Order
Laplacian
15 Vertical Mixing
2 km 60 uneven(100,2,.5)
SDJ Pacanowski-Philander
2nd Order
Laplacian
16 Vertical Mixing
2 km 60 uneven(100,2,.5)
SDJ LOAM 2nd Order
Laplacian
17 Vertical Mixing
2 km 60 uneven(100,2,.5)
SDJ LMD without BKPP
2nd Order
Laplacian
18 Vertical Mixing
2 km 60 uneven(100,2,.5)
SDJ LMD modified 2nd Order
Laplacian
19 Vertical Mixing
2 km 60 uneven(100,2,.5)
SDJ Generic Length Scale
2nd Order
Laplacian
20 Horizontal Mixing
2 km 60 uneven(100,2,.5)
SDJ LMD 1 m2 s-1
21 Horizontal Mixing
2 km 60 uneven(100,2,.5)
SDJ LMD 1x10-6
m2 s-1
22 Other 2 km 60 uneven(100,2,.5)
SDJ LMD 2nd Order
Laplacian
Latitude Shift
5oS
Inverse Richardson No.
-4000
-2000
0
Dep
th (
m)
0.01
0.10
1.00
4.00
Bottom
-4000
-2000
0
Dep
th (
m)
-4000
-2000
0
Dep
th (
m)
-4000
-2000
0
Dep
th (
m)
32.0 32.2 32.4 32.6 32.8Latitude
-4000
-2000
0
Dep
th (
m)
128o 1.7' W
127o 51.4'W
127o 41.0' W
127o 30.7'W
127o 20.4'W
Base Case (1)
