Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations CeSOS Seminar 19 December 2005
Bjørn Gjevik
Department of Mathematics, University of Oslo
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.1/?? • Eddy formation in shelf slope currents
• Some aspects of tsunami modelling
Plan for presentation
• High resolution models of tidal currents in coastal waters
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.2/?? • Some aspects of tsunami modelling
Plan for presentation
• High resolution models of tidal currents in coastal waters
• Eddy formation in shelf slope currents
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.2/?? Plan for presentation
• High resolution models of tidal currents in coastal waters
• Eddy formation in shelf slope currents
• Some aspects of tsunami modelling
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.2/?? Depth integrated eqs. of motion
∂U ∂ U 2 ∂ UV ∂η √U 2 + V 2 U + ( )+ ( ) fV = gH cD +Bx ∂t ∂x H ∂y H − − ∂x− H H ∂V ∂ UV ∂ V 2 ∂η √U 2 + V 2 V + ( )+ ( )+fU = gH cD +By ∂t ∂x H ∂y H − ∂y − H H
(U, V ) horizontal volume fluxes
H = H0 + η total depth (undisturbed + displacement)
cD bottom friction coefficient (quadratic)
(Bx, By) horizontal eddy viscosity (Smagorinski) g acceleration of gravity f Coriolis parameter
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.3/?? Smagorinski turbulence model (LES) 2 2 Bx = ν U By = ν V 5 5 with eddy viscosity
2 ∂u 2 1 ∂u ∂v 2 ∂v 2 1 ν = ql [( ) + ( + ) + ( ) ] 2 ∂x 2 ∂y ∂x ∂y where depth mean current U V u = , v = H H q = constant O(0.1) – O(1), l = length scale O(∆x) ν = O( 10 m2/s)
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.4/?? Continuity equation
∂η ∂U ∂V = ∂t − ∂x − ∂y
(U, V ) horizontal volume fluxes η sea surface displacement
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.5/?? • Topographic Rossby waves
• Quasi-geostrophic fl ow
Classic analytical solutions
• Kelvin waves
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.6/?? • Quasi-geostrophic fl ow
Classic analytical solutions
• Kelvin waves
• Topographic Rossby waves
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.6/?? Classic analytical solutions
• Kelvin waves
• Topographic Rossby waves
• Quasi-geostrophic fl ow
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.6/?? • Space staggered B and C-grids
• Experiments with optimalization (adjoint models)
Numerics
• Finite differencing schemes
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.7/?? • Experiments with optimalization (adjoint models)
Numerics
• Finite differencing schemes
• Space staggered B and C-grids
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.7/?? Numerics
• Finite differencing schemes
• Space staggered B and C-grids
• Experiments with optimalization (adjoint models)
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.7/?? The Nordic Seas
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.8/?? Hierarchy of model domains
30' 78 o N SPITSBERGEN
65oN
72 o N Rørvik
30'
66 o N 64oN
60 o 30' N Trondheim NORWAY
63oN 54 o N GREAT− BRITAIN 30' Ålesund
10 o o E W 40 o o o 0 E 62 N o o 30 o o 10 E 20 E 6 E 8oE 10oE 12 E
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.9/?? Lofoten Northern Norway (M2)
338 332 326 320 314 308 302
350 62 ∆x = 500 m
A64 56 Sea level (cm) 300 62 58 Stasjon Obs. Mod. 60 250 H N Bodø 86.9 86.1 200 62 T Narvik 99.3 99.6 90 km Kabelvåg 92.6 91.3 88 64 86 150 332 70 Tangstad 62.3 63.7 84 3000 66 2500 82 2000 Harstad 69.3 67.5 100 68 Bodø 1500 1000 Andenes 64.8 63.0
80 500 78
326 70 300 250 72 76 50 74 200 150 100
land 0 0 50 100 150 200 250 300 350 400
km Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.10/?? Model domain I
Harstad
100 Horizontal grid: Hinnøya ∆x = 50-100 m Evenskjær 80 Narvik
60
40
20
0 Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.11/?? 0 20 40 60 80 km Model domain II
25 ∆x =
HINNØYA 25-50 m 20 Steinsland
15 Evenskjæ r Ballstad Sandtorg km
10 300 200 150 TJELDØYA 100 Ramsund 75 50 5 40 30 20 10
land 0 Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.12/?? 0 5 10 15 20 25 30 km SANDTORGSTRAUMEN t = 1 hour
16
RUN65 (M2)
15
14 km
13 Sandtorg
12 Fjelldal
19 20 21 22 Modelling23 tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.13/??
km 0.5 m/s SANDTORGSTRAUMEN t = 2 hour
16
RUN65 (M2)
15
14 km
13 Sandtorg
12 Fjelldal
19 20 21 22 Modelling23 tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.14/??
km 0.5 m/s SANDTORGSTRAUMEN t = 3 hours
16
RUN65 (M2)
15
14 km
13 Sandtorg
12 Fjelldal
19 20 21 22 Modelling23 tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.15/??
km 0.5 m/s SANDTORGSTRAUMEN (t = 4 hours)
16
RUN65 (M2)
15
14 km
13 Sandtorg
12 Fjelldal
19 20 21 22 Modelling23 tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.16/??
km 0.5 m/s SANDTORGSTRAUMEN t = 5 hours
16
RUN65 (M2)
15
14 km
13 Sandtorg
12 Fjelldal
19 20 21 22 Modelling23 tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.17/??
km 0.5 m/s SANDTORGSTRAUMEN t = 6 hours
16
RUN65 (M2)
15
14 km
13 Sandtorg
12 Fjelldal
19 20 21 22 Modelling23 tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.18/??
km 0.5 m/s SANDTORGSTRAUMEN t = 7 hours
16
RUN65 (M2)
15
14 km
13 Sandtorg
12 Fjelldal
19 20 21 22 Modelling23 tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.19/??
km 0.5 m/s SANDTORGSTRAUMEN t = 8 hours
16
RUN65 (M2)
15
14 km
13 Sandtorg
12 Fjelldal
19 20 21 22 Modelling23 tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.20/??
km 0.5 m/s SANDTORGSTRAUMEN t = 9 hours
16
RUN65 (M2)
15
14 km
13 Sandtorg
12 Fjelldal
19 20 21 22 Modelling23 tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.21/??
km 0.5 m/s SANDTORGSTRAUMEN t = 10 hours
16
RUN65 (M2)
15
14 km
13 Sandtorg
12 Fjelldal
19 20 21 22 Modelling23 tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.22/??
km 0.5 m/s SANDTORGSTRAUMEN t = 11 hours
16
RUN65 (M2)
15
14 km
13 Sandtorg
12 Fjelldal
19 20 21 22 Modelling23 tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.23/??
km 0.5 m/s SANDTORGSTRAUMEN t = 12 hours
16
RUN65 (M2)
15
14 km
13 Sandtorg
12 Fjelldal
19 20 21 22 Modelling23 tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.24/??
km 0.5 m/s SANDTORGSTRAUMEN t = 13 hours
16
RUN65 (M2)
15
14 km
13 Sandtorg
12 Fjelldal
19 20 21 22 Modelling23 tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.25/??
km 0.5 m/s Harmonic analysis (station 19)
0.7
0.6
0.5
0.4
0.3 Current (m/s) 0.2
0.1
0 20 30 40 50 60 70 80 90 100 110 Time (hours)
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.26/?? STEINSLANDSTRAUMEN (HW)
22
RUN65 Mean tide (M2) 21 Max current 2 m/s Hinnøya
20 Tjeldsundbrua km
19 Steinsland
18
21 22 23 24 Modelling25tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.27/??
km 0.5 m/s BALLSTADSTRAUMEN (HW)
13.0
12.5
Ballstad 12.0 km 11.5
RUN65 11.0 Mean tide (M2) Max current 1.0 m/s
10.5 Tjeldøya
10.0 10.5 11.0 11.5 12.0 Modelling12.5tidal currents13.0, shelf slope eddies13.5, and tsunamis with14.0shallow water equations14.5 – p.28/??
km 0.5 m/s RAMSUNDET (HW)
9
RUN65 Mean tide (M2) 8 Max current1 m/s
7 km Sandbogstraumen
6
Orlogsstasjon
5
18 19 20 21 Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.29/??
km 0.5 m/s On a rock in Tjeldsundet
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.30/?? Tidal currents in navigation charts
Storfosna Garten
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.31/?? The Norwegian shelf
70Ê
65Ê
2000 1000 200 SWEDEN
500 1500
NORWAY 60Ê Oslo
55Ê
ENGLAND GERMANY
0Ê 5Ê 10Ê 15Ê Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.32/??
350Ê 355Ê NDP measurement programme
68.0
2000
Gjallar Vema/Nyk1400 200 1000 67.0
3000
750
66.0 300
200
2000 3000 Helland-Hansen
1400 300 65.0 200
500 300
64.0 Ormen Lange Mùre West
1000 200
63.0 750
62.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.33/?? Current profile Svinoy section
From Orvik and Mork (1996)
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.34/?? Eddies seen from space
ERS-2 SAR Image 19.07.87, 10:59 UTC From K. Kloster, NERSC, Bergen
Between Ormen Lange and Helland Hansen
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.35/?? Idealized shelf slope model
x u¯ y
Barotropic
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.36/?? Eddy formation: Ormen Lange
0.5 m/s 600 0.5 m/s 0.5 m/s
550
180 180 500
160 160 450
400 140 140
350 120 120
300 km 100 100 km km
250 80 80
200
60 60 150
40 40 100
50 20 20
0 0 0 0 50 100 150 200 250 300 140 160 180 200 220 140 160 180 200 220 km km km Inflow jet 7.5 days 10 days
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.37/?? Stability analysis of steady current
x u¯ u = u¯ + uˆ(y) cos(kx ωt) y v = vˆ(y) sin(kx −ωt) η = η¯ + ηˆ(y) cos(kx − ωt) −
u¯ mean along shelf current η¯ mean sea surface displacement k wave number ω angular velocity uˆ,vˆ,ηˆ perturbations
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.38/?? Linearized eqs. for perturbations
uk¯ , h¯k, h¯D Dh¯ ηˆ ηˆ − − gk, u¯k, f Du¯ uˆ = ω uˆ − gD, f, uk¯ vˆ vˆ
D d ≡ dy The eigenvalue problem for ω is solved by discretizing on a spaced staggered grid using NAG-routines.
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.39/?? Dispersion diagram
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.40/?? Eigen-mode: Ormen Lange profile
1 m/s
180
160 λ = 43 km, T = 33.7 hours Mean current added 140
120
100 km
80
60
40
20
0 140 160 180 200 220 km
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.41/?? Tsunami in Tafjord April 1934
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.42/?? Run-up along the fjord
SYLTE 3.5 4.3 10.6 9.4 5.7 6.6 7.7 12.0 10.0 Muri 12.0 12.3 10.3 10.7 12.6
Lingås 10.5 Hones Haugs- Fjöra N Linge bukt
Vinsnes Oksneset 10.8 Osvik 4.0 TAFJORDEN 24.4 3.0 4.6 21.2 12.5 12.6 2.8 6.4 16.0 av bølger Ingen spor 7.5 Skred 2.8 KARLSTAD Alvika r e 31.4 k r 32.4 62.3 e 32.7 KORSNES m
n
e
g 23.9
n 13.6 I 37.3 17.4 16.0
1.0 23.6 NORDDAL MULDAL Seineset 17.1 12.9 ~ 32 16.1 Södals- 17.8 15.7 vik 19.9 16.8 Skjegg- 24.8 16.3 Gjeit- vika hammer 27.0 19.5 25.5 33.8 15.8 Meter 5.0 10.9 10.3 0 1000 2000 5.9 5.3 4.5 8.0 5.7 10.1 7.2 13.1 3.6 11.7 11.9 13.6 15.7 14.6 TAFJORD
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.43/?? 2) Propagation over open ocean 3) Amplification and run-up at the coast
For modelling and predictions points 1) and 3) are the greatest challenges
Three phases of a tunamis event
1) Generation of waves in the source area
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.44/?? 3) Amplification and run-up at the coast
For modelling and predictions points 1) and 3) are the greatest challenges
Three phases of a tunamis event
1) Generation of waves in the source area 2) Propagation over open ocean
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.44/?? For modelling and predictions points 1) and 3) are the greatest challenges
Three phases of a tunamis event
1) Generation of waves in the source area 2) Propagation over open ocean 3) Amplification and run-up at the coast
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.44/?? Three phases of a tunamis event
1) Generation of waves in the source area 2) Propagation over open ocean 3) Amplification and run-up at the coast
For modelling and predictions points 1) and 3) are the greatest challenges
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.44/?? Model equations (2-D Boussinesq)
∂U ∂ U 2 ∂η h2 ∂3U U U + ( ) = gh + cD | | ∂t ∂x h − ∂x 3 ∂t∂x2 − h2 ∂η ∂U ∂b = o ∂t − ∂x − ∂t
U Horizontal volume flux
h = h0 + b0 + η Water depth (mean + changes) η Displacement of sea surface
bo Sea bed displacement
cD Bottom friction coefficient g Acceleration of gravity
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.45/?? Waves from rock slides in fjords
Pedersen og Johnsgard 1996
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.46/?? • Non hydrostatic effects
• Wave-current interaction
Extention of shallow water models
• Vertical current profile by ’2.5-D models’
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.47/?? • Wave-current interaction
Extention of shallow water models
• Vertical current profile by ’2.5-D models’
• Non hydrostatic effects
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.47/?? Extention of shallow water models
• Vertical current profile by ’2.5-D models’
• Non hydrostatic effects
• Wave-current interaction
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.47/?? Vertical current profiles from
∂u ∂η ∂ ∂u fv = g + (ν ) ∂t − − ∂x ∂z ∂z ∂v ∂η ∂ ∂v fu = g + (ν ) ∂t − − ∂y ∂z ∂z
η, and U = udz R known from depth integrated model
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.48/?? Publications by Oslo group (I)
Gjevik et al.(1994) Model simulations of the tides in the Barents • Sea. J. Geophys. Res., Vol 99, No C2, side 3337–3350. Moe et al. (2002) A high resolution tidal model for the area around • the Lofoten Islands. Cont. Shelf. Res., Vol 22, 485-504. Ommundsen (2002) Models of cross shelf transport introduced by • the Lofoten Maelstrom. Cont. Shelf. Res., Vol 22, 93-113. Moe et al. (2003) A high resolution tidal model for the coast of • Møre and Trøndelag. Norwegian J. Geography, Vol 57, 65-82. Gjevik et al. (2004) Implementation of high resolution tidal current • field in electronic navigational chart systems. Preprint Series. Dept of Math., UiO ISSN 0809-4403. (J. Marine Geodesy, in press)
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.49/?? Publications (II)
Moe et al. (2003) A high resolution tidal model for the coast of • Møre and Trøndelag. Norwegian J. Geography, Vol 57, 65-82. Gjevik et al. (2004) Implementation of high resolution tidal current • field in electronic navigational chart systems. J. Marine Geodesy (in press) Hjelmervik et al (2005) Implementation of Non-Linear Advection • Terms in a High Resolution Tidal Model. Preprint Series, Dept. of Math., University of Oslo. No. 1, ISSN: 0809-4403. Gjevik et al. (2002) Idealized model simulations of barotropic flow • on the Catalan shelf. Continental Shelf Research, 22, 173-198. B. Gjevik (2002) Unstable and neutrally stable modes in • barotropic and baroclinic shelf slope currents. Preprint Series, Dept. of Math., Univ. of Oslo, No 1. Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.50/?? Publications (III)
Harbitz, (1992) Model simulations of tsunamis generated by the • Storegga slides. Marine Geology, Vol. 105, 1–21. Harbitz, et al. (1993) Numerical simulations of large water waves • due to landslides. J. Hydraulic Engineering, Vol. 119, No. 12, side 1325–1342. Johnsgard and Pedersen (1997) A numerical model for • three-dimensional run-up. Int. Num. Meth. Fluids 24, 913–931. Pedersen (2005) Modeling run-up with depth integrated equation • models. In Advances in Coastal and Ocean Engineering. World Scientific Publishing. Eds. Yeh and Synolakis. Langtangen and Pedersen (2002) Propagation of large • destructive waves. J. Appl. Mech. Engineering. 7, 1, 187–204.
Modelling tidal currents, shelf slope eddies, and tsunamis with shallow water equations – p.51/??