open challenges in ocean modelling with focus on the
TRANSCRIPT
Open Challenges in Ocean Modelling withfocus on the Nordic Seas and non-hydrostatic
pressure
Jarle Berntsen
Department of Mathematics, University of Bergen
2008 Princeton University
Jarle Berntsen Overview
Flow and processes in the Nordic Seas
History and background
The general flow
The role of small scale events
Feedback from the smaller scale processes to larger scaleflow
Non-hydrostatic pressure
Jarle Berntsen Overview
General Circulation in The Nordic Seas - 1887 view
Circulation in the Nordic Seas (From Mohn 1887)
Jarle Berntsen Overview
General Circulation in The Nordic Seas - 1909 viewCirculation in the Nordic Seas (From Helland-Hansen andNansen 1909)
Jarle Berntsen Overview
General Circulation in The Nordic Seas
Present view of the circulation in the Nordic Seas (From theGeophysical Institute, UoB )
Jarle Berntsen Overview
Why is this flow important?
Climate
Transport of heat and nutrients to our waters
Eco-system
Fisheries
Coral Reefs
Fish farming
Off-shore industrial activities
Dispersal of pollution ++
A number of disciplines, institutes, organisations,governmental bodies involved
Jarle Berntsen Overview
Good cod recruitment depend on warm years
From Helland-Hansen and Nansen, The Norwegian Sea, 1909
Jarle Berntsen Overview
Coral reefs along the shelf
From Fosså, Mortensen and Furevik 2002
Jarle Berntsen Overview
Artificial Upwelling in Fjords
From Aure et al. 2007
Jarle Berntsen Overview
The Nordic Seas
(From Norsk Hydro + Bjørn Gjevik)
Jarle Berntsen Overview
The Ormen Lange area
(From Norsk Hydro)
Jarle Berntsen Overview
The Storegga area
(From Norsk Hydro)
Jarle Berntsen Overview
Deep water industrial activity
(From Norsk Hydro)
Jarle Berntsen Overview
Local topography in the shelf slope
(From Norsk Hydro)
Jarle Berntsen Overview
What do we need to know?
For the general understanding and for climate studies:Mean properties
General circulation pattern and long time variability
Variability (year to year and longer) of inflow of AtlanticWater and heat
For local processes and industrial purposes: Also local,rapid, and strong events
Transfer of energy towards smaller scale/high frequencyevents
The small scale events also affect the general circulation
The role of interactions between stratified flow andtopography on the oceanic mixing
Processes on length scales from global scale down toapprox. 1 mm
Jarle Berntsen Overview
Methods
Measurements - frequency and spatial distribution?
Laboratory experiments including rotating tables
Simplified process oriented mathematical models
Numerical ocean models based on the Reynolds averagedNavier-Stokes equations (RANS)
RANS models plus data assimilation
In RANS modelling: The fundamental equations are wellknown, but the results are still very sensitive to the choice ofsub-grid scale closure
Jarle Berntsen Overview
Observations in the Ormen Lange area
(From Fugro Oceanor )
Jarle Berntsen Overview
Cross section of salinity
(From Fugro Oceanor )
Jarle Berntsen Overview
Current events
(From Fugro Oceanor )
Jarle Berntsen Overview
Current events
(From Vikebø, Berntsen and Furnes 2004 )
Jarle Berntsen Overview
Modelled current events
(From Vikebø, Berntsen and Furnes 2004 )
Jarle Berntsen Overview
Downwelling along the slope
(From Fugro Oceanor )
Jarle Berntsen Overview
Why is it difficult to capture the rapid events withocean models?
Grid sizes in the range from 20 km to 200 m used
Still unresolved processes
Nesting does not help
A lack of good quality initial data and boundary values
Climatologies very useful
Climatologies lack fronts and interfaces
Strong events often associated with the movements of theinterfaces and fronts
Process oriented studies required to understand the events
Jarle Berntsen Overview
Observations at TH11W
(From Fugro Oceanor )
Jarle Berntsen Overview
Modelled solitary wave train
(From Berntsen, Mathisen, and Furnes 2007 )
Jarle Berntsen Overview
Equation for the surface elevation
Airy function solution for simplified model equations
η(x , t) ∼1
2Ct1/3Ai(
x −√
ghCt1/3
)
Jarle Berntsen Overview
Solitary wave trains
Driving forces - wind and tide
Where can we find them?
3D studies difficult due to resolution issues
The use of 2D (x , y) model equations to locate ’hot spots’
Subsequent measurements
In the future: Full 3D simulations
Jarle Berntsen Overview
The Navier-Stokes equations
ρDuDt
= −∇p + ρg + µ∇2u
For an incompressible fluid. Also conservation of temperatureand heat and an equation of state.
Jarle Berntsen Overview
The Reynolds averaged Navier-Stokes equations
u = u + u′
DuDt
= −1ρ0
∇p +ρ
ρ0g + ∇ · (AE∇u)
Buossinesq + Hydrostatic
The Closure problem?
Also transfer of energy from small scale events to largescale flow
How to parameterize effects of sub-grid scale topography?
Jarle Berntsen Overview
Vertical coordinate?
(From the DYNAMO report 1997 )
Jarle Berntsen Overview
Non-hydrostatic pressure in σ-models
Status on non-hydrostatic ocean models
The non-hydrostatic pressure estimation in σ-coordinatemodels
When do we need to include non-hydrostatic pressureeffects?
Tidal flow over a sill
Hydrostatic versus non-hydrostatic results as functions ofgrid size and SGS closure
Jarle Berntsen Overview
Non-hydrostatic ocean models
In z-coordinates: The MITgcm
In σ-coordinates: Kanarska and Maderich (2003) - POM
In ROMS: Kanarska, Shchepetkin, and McWilliams (2007)
Also available in other models
May be expensive to compute non-hydrostatic pressure
Jarle Berntsen Overview
Non-hydrostatic pressure in σ-models
After transformation from z-coordinates (KM03) :
∂
∂x
(
D∂P∂x
)
−∂
∂x
(
∂P∂σ
(
σ∂D∂x
+∂η
∂x
))
+1D
∂2P∂2σ
−
∂
∂σ
(
∂P∂x
(
σ∂D∂X
+∂η
∂x
))
+1D
∂
∂σ
(
∂P∂σ
(
σ∂D∂x
+∂η
∂x
)2)
=
ρ0
∆t
(
∂UD∂x
+∂ω
∂σ+
∂η
∂t
)
.
Jarle Berntsen Overview
Non-hydrostatic pressure in σ-models
Simplified version in Berntsen and Furnes (2005) :
∂
∂x
(
D∂P∂x
)
+1D
∂2P∂2σ
=ρ0
∆t
(
∂UD∂x
+∂ω
∂σ+
∂η
∂t
)
.
5 versus 9 points involved in 2D studies7 versus 19 points involved in 3D studiesTotal errors = Model errors + Numerical errors
Jarle Berntsen Overview
Surface boundary condition for P?
P = 0 in KM03∂P∂n = 0 in MITgcm
Also mixed BCs suggested
Jarle Berntsen Overview
When do we need non-hydrostatic pressure?
Scale analysis- non-hydrostatic parameter
1 to 10 km suggested in Marshall et al. 1997
Run time decisions suggested in Wadzuk and Hodges 2004
Process oriented studies
Sensitive to SGS parameterisation
If we allow PNH effects without resolving the realnon-hydrostatic processes, we may get strong aliasingeffects
Jarle Berntsen Overview
Super-critical flow over a sill (Loch Etive)
See Inall et al. 2004,2005, Stashchuk et al. 2007, and severalpapers by Xing and Davies.XD demonstrated strong non-hydrostatic effects using theMITgcm and ∆x = 12.5m.
Temperature field after 1/4 T with ∆x = 12.5 m.
8.5
9
9
9.5
9.5
9.5
10
10
10
10
10.5
10.5
10.5
11
11
11
11
11.5
11.5 11.5
11.5
11.5
12
12
12
12
1212
12
12
12
12 12
12
12
12.5
13
T (ci=0.25 C)
distance [m]
dept
h [m
]
0 200 400 600 800 1000 1200 1400 1600 1800 2000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Jarle Berntsen Overview
Pressures at maximum inflow
0 100 200 300 400 500 600 700 800 900 1000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Distance from the top of the sill [m]
Dep
th [m
]
0
0
0
00
0
0
0
00
00
2
2
2
2
2
22 2
2
4
4
4
4
4 44
4
6
6
6
6
8
8
8
8
8
10
10
10
1214P (ci=2.0 N/m**2)
0
0 0
0 00
00
00
00
00
−2
−2
−2
−2
−2
−2
−2 −
2
−2
−2
−2
−4
−4
−4
−4
−4
−4 −
4
−4
−4
−4
−6
−6
−6
−6
−8
−8
−8
−8
−8−1
0
−10
0 100 200 300 400 500 600 700 800 900 1000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Distance from the top of the sill [m]
Dep
th [m
]
0
0
0
00
0
0
00
0
2
2
2
2
2
4
4
4
4
6
6
6
6
8
8
8
8
10
10
10
10
12
12
12
12
14
14
14
14
16
16
16
18
18
18
20 2022
22
24
24
26P (ci=2.0 N/m**2)
0
0
0 0
0
0
0
0 0
0
00
0
−2
−2
−2
−2
−2
−2
−2
−2
−4
−4
−4
−4−
4
−4
−4
−6−
6
−6 −
6
−6
−8
−8
−8
−8
−10
−10
−10
−10
−12
−12
−12 −
12
−14
−14
−14
−16
−16
−16
−18
−20
−22
0 100 200 300 400 500 600 700 800 900 1000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Distance from the top of the sill [m]
Dep
th [m
]
0
0 0
00 0
0
00
0
0
0
2
22 2
2
22
44
4
44
4
6
6
6
6
6
8
8
8
8
8
10
10
10
12
12 1416
P (ci=2.0 N/m**2)
0
0 0
00 0
0
00
00
0−2
−2
−2
−2
−2
−2
−2
−2 −2
−2
−4
−4 −4−
4
−4
−4 −
4
−4
−4
−4
−6
−6
−6
−6
−6
−6
−8
−8
−8
−8
−8
−10
−10 −
12
∂P/∂n = 0 at the surface
0 100 200 300 400 500 600 700 800 900 1000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Distance from the top of the sill [m]D
epth
[m]
0 0
0
0
00
0
0
0
00
2
2 2
2
2
4
4
4
4
4
6
6
6
6
6
8
8
8
8
10
10
10
12
12
14
14
1618
20
22
24
26
P (ci=2.0 N/m**2)
0 0
0
0
0
0
0
0
0
00
−2
−2
−2
−2
−2
−2
−2
−2
−2
−4
−4
−4
−4
−4
−4
−4 −
4
−6
−6
−6 −
6
−6
−8
−8
−8
−8
−10
−10
−10
−10
−12
−12
−12
−14
−14
−16
−16
−18
P = 0 at the surface
Upper panel KM-method and lower panel BF-method
Jarle Berntsen Overview
Densities at maximum inflow
0 100 200 300 400 500 600 700 800 900 1000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Distance from the top of the sill [m]
Dep
th [m
]
24.2
24.2
24.2
24.2 24
.2
24.2
5
24.25 24.3
24.3
524
.424
.45
24.5
24.7
5
RHO (ci=0.05 )
0 100 200 300 400 500 600 700 800 900 1000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Distance from the top of the sill [m]
Dep
th [m
]
24.15
24.2
24.2
24.2
24.2
5
24.2524.3
24.35
24.4
24.6
RHO (ci=0.05 )
0 100 200 300 400 500 600 700 800 900 1000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Distance from the top of the sill [m]
Dep
th [m
]
24.15
24.2
24.2
24.2
5
24.3
24.3
524.424.45 24.5
24.7
RHO (ci=0.05 )
∂P/∂n = 0 at the surface
0 100 200 300 400 500 600 700 800 900 1000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Distance from the top of the sill [m]D
epth
[m]
24.2
24.2
24.224.2
5
24.25
24.3
24.35 24.424.4524.5
24.6
RHO (ci=0.05 )
P = 0 at the surface
Upper panel KM-method and lower panel BF-method(From paper submitted with Keilegavlen)
Jarle Berntsen Overview
Temperatures at maximum inflow for ∆x = 100 m
0 200 400 600 800 1000 1200 1400 1600 1800 2000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
distance [m]
dept
h [m
]
8.258.5
8.759
9.259.59.751010.25 10.5
10.75
11 11
11.25
11.25
11.5
11.5
11.75
11.7511.75
12
12 12
12
12.25
12.2
5
12.25
12.5
12.5
12.7513 13
T (ci=0.25 C)
0 200 400 600 800 1000 1200 1400 1600 1800 2000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
distance [m]
dept
h [m
]
8.258.58.7599.25
9.59.751010.2510.5
10.75
11
11
1111.25
11.2
5
11.5
11.5
11.75
11.75
11.75
12
12 12
12
12.25
12.2
5
12.25
12.512.512.7513 13
T (ci=0.25 C)
Non-hydrostatic results on top and hydrostatic results bel ow
Jarle Berntsen Overview
Temperatures at maximum inflow for ∆x = 12.5 m
0 200 400 600 800 1000 1200 1400 1600 1800 2000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
distance [m]
dept
h [m
]
8.258.58.75
8.759
9.259.5
9.7510
10
10.25
10.2
5
10.5
10.5
10.5
10.75
10.7
5 10.7
5
10.75
11
11
11
11 11
11.25
11.25 11.25 11.2
5
11.2511.5
11.5
11.5
11.5
11.5
11.5
11.5
11.75
11.7
5
11.7
5
11.75 11.7
5
11.7511.75
11.7
5
11.7511.7
5 12
12
12 12 12 12
12
12
12
12
12
12
12
12
1212
12
12
12.2
5
12.25
12.2
5
12.2512.25
12.2
5
12.5
12.7513
T (ci=0.25 C)
0 200 400 600 800 1000 1200 1400 1600 1800 2000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
distance [m]
dept
h [m
]
8.258.5
8.75 8.759
9
9.25
9.259.5 9.59.75
9.75
9.7510
10 10
10.25 10.25 10.2510.5
10.5
10.5
10.75
10.75 10.7
5
10.7
5
10.75
11
1111 11
11
11.25
11.2
511.25
11.25
11.25
11.5
11.5
11.5
11.5
11.5
11.511.75
11.75
11.7
5
11.75 11.7
5 11.75
11.7
512
12
12
12
12 12
12
12
12.25
12.25
12.25
12.2
5
12.2512
.5
12.5
12.75 12.75
13 13
T (ci=0.25 C)
Non-hydrostatic results on top and hydrostatic results bel ow
Jarle Berntsen Overview
Temperatures at maximum inflow for ∆x = 1.5625m
0 200 400 600 800 1000 1200 1400 1600 1800 2000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
distance [m]
dept
h [m
]
8.258.5
8.75 8.759 9
9.25 9.259.59.5
9.75 9.75
9.7510
10
10
10.2
5 10.25
10.2
5
10.5
10.5 10.5
10.75
10.7
5
10.7
5
10.7
5
11
11
11
11
11
11
11
11
11
11
11
1111.2
511.25
11.2
5
11.2
5
11.25
11.2511.25
11.2
5
11.2
5
11.2511.2
5
11.25
11.25
11.5
11.5
11.5
11.5
11.5
11.5
11.511.5
11.5
11.5
11.5
11.5
11.5
11.5
11.5
11.5
11.5
11.5
11.511.75
11.7
5
11.75
11.7
5
11.75
11.7
5 11.75 11.75
11.7511.7511.75
11.7
5
11.7
511.75
11.7511.75
11.75
11.7
511
.7512
12
1212
12
12
12
12
1212
12 12 12
12
1212 12
12
12
12
12
12
12
12
12.2
5
12.25
12.2
5
12.2
512
.25
12.25
12.2
512
.25
12.2
5
12.2
5
12.2
5
12.2512.25
12.25 12.2
5
12.25
12.25
12.25
12.2512.25
12.512.5
12.5
12.5
12.5
12.5
12.512
.5
12.5
12.5
12.5
12.5
12.5
12.512.5
12.7512.75
12.7
5
13
13
T (ci=0.25 C)
0 200 400 600 800 1000 1200 1400 1600 1800 2000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
distance [m]
dept
h [m
]
8.258.5
8.75 8.759 9
9.259.25
9.5 9.5 9.59.75
9.75 9.7510 10 101010.25
10.25
10.25 10.2510.5
10.5
10.5 10.5
10.75
10.75 10.7
5
10.7
5
10.7511
11 11
11 11 11
11
11
11
11
11
11 1111
11.25
11.2511.25
11.25 11.2
5 11.25 11.25
11.2511.2
5
11.25 11.25
11.2
5
11.2
511
.25
11.5 11.5
11.5
11.5
11.5
11.5
11.5
11.5
11.5
11.511.511.511.5
11.5
11.511.5
11.75
11.7
5
11.75
11.7
5 11.7
511
.75
11.75
11.75 11.75 11.75 11.75 11.7
511.75
11.7511.7511
.75
11.75
11.75
11.75
11.75
11.7
5
11.75
11.7
5
11.7511.7511.7511.75
11.7
5
11.7
511.75
12
12
12
12 12
12 12
12
12
121212
12
12.25 12.25
12.25
12.25 12.25
12.25
12.5
12.5 12.5 12.5
12.5
12.75 12.7512.75
12.75
13 13 13
13
T (ci=0.25 C)
Non-hydrostatic results on top and hydrostatic results bel owJarle Berntsen Overview
Time mean temperatures for ∆x = 1.5625 m
0 200 400 600 800 1000 1200 1400 1600 1800 2000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
distance [m]
dept
h [m
]
8.258.5
8.759 99.25 9.259.59.59.75
9.75101010.2510.2510.5
10.510.75
10.75
11
11
11
11.25
11.25
11.25
11.5
11.5
11.5
11.75
11.7511.75
12
12
12
12
12.25
12.25
12.5 12.5
12.5
12.75 12.7513
T (ci=0.25 C)
0 200 400 600 800 1000 1200 1400 1600 1800 2000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
distance [m]
dept
h [m
]
8.258.5
8.759 9
9.25 9.259.5 9.59.75 9.7510 1010.2510.5
10.510.7510.75
11
1111
11.25
11.2511.25
11.5
11.511.5
11.75
11.7511.75
12
12
12.25
12.25
12.5
12.5
12.75
12.75
1313
T (ci=0.25 C)
Non-hydrostatic results on top and hydrostatic results bel ow
Jarle Berntsen Overview
Time mean U-velocities for ∆x = 1.5625 m
0 200 400 600 800 1000 1200 1400 1600 1800 2000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
distance [m]
dept
h [m
]
0
0
00
10
10
10
20
20
30 3040
405060
8090
U (ci=10 cm/s)
00
0
0 0
−10
0 200 400 600 800 1000 1200 1400 1600 1800 2000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
distance [m]
dept
h [m
]
0
0
0
10
10
20
20
3030405060
90
U (ci=10 cm/s)
0
0
0
Non-hydrostatic results on top and hydrostatic results bel ow
Jarle Berntsen Overview
Time mean U-velocities
0 200 400 600 800 1000 1200 1400 1600 1800 2000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
distance [m]
dept
h [m
]
0
0
00
10
10
10
20
20
30 3040
405060
8090
U (ci=10 cm/s)
00
0
0 0
−10
0 200 400 600 800 1000 1200 1400 1600 1800 2000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
distance [m]
dept
h [m
]
0
0
0
10
10
20
40
U (ci=10 cm/s)
0
0
00
−10
Non-hydrostatic results for ∆x = 1.5625 m on top and for ∆x = 100 m below
Jarle Berntsen Overview
Measuring the effects in the 2-norm
‖ φH − φNH ‖=
√
1Area
∫ 2000m
0m
∫ η(x,t)
H(x)(φH − φNH)2dxdz
2-norms also computed for the time mean fields
Jarle Berntsen Overview
2-norms of differences in temperatures
10 20 30 40 50 60 70 80 90 1000
0.1
0.2
0.3
0.4
0.5
DXN
orm
of Θ
H −
ΘN
H [C
]
10 20 30 40 50 60 70 80 90 1000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
DX
Nor
m o
f ΘH −
ΘN
H [C
]
Results at max. inflow on top and for time mean fields belowSquares used for low visc./diff. exp.Circles used for large visc./diff. exp.
Jarle Berntsen Overview
2-norms of differences in U-velocities
10 20 30 40 50 60 70 80 90 1000
0.05
0.1
0.15
0.2
0.25
DX
Nor
m o
f UH −
UN
H [m
/s]
10 20 30 40 50 60 70 80 90 1000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
DX
Nor
m o
f UH −
UN
H [m
/s]
Results at max. inflow on top and for time mean fields belowSquares used for low visc./diff. exp.Circles used for large visc./diff. exp.
Jarle Berntsen Overview
Sensitivity to the slope
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06
0.08
0.1
0.12
SLOPE
Err
ors
[m/s
]
Errors in time average U as function of slope.Squares used for errors from non-hydrostatic P errors
Circles used for errors from internal pressure gradient err orsDiamonds used for ’errors’ due to uncertainties in the SGS cl osure
Jarle Berntsen Overview
Discussion-I
The primary model fields in non-hydrostatic σ-coordinatemodels robust to the choice of method for the pressure, andto the surface boundary condition
If viscosity/diffusivity small, stronger non-hydrostatic effectswith smaller ∆x
Also clear non-hydrostatic effects on the time mean fields
For the sill case, the turbulent rotor in the lee is missing inthe hydrostatic results
For the sill case, small non-hydrostatic effects with ∆xlarger than 25 m
Possible aliasing effects with too large ∆x
Consequences for 3D studies
Jarle Berntsen Overview
Discussion-II
For full non-hydrostatic effects: DNS
Internal waves + Kelvin-Helmholtz instabilities represented
Parameterizations of non-hydrostatic effects into larger gridsize hydrostatic models?
How shall we include the effects of the lacking counterpressure in the lee of the sill?
There is a cost involved when computing thenon-hydrostatic pressure
We need an elliptic pressure solver
10 times SOR per time step
Jarle Berntsen Overview