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

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 (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 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 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

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/??