Numerical studies of dispersion due to tidal flow through Moskstraumen, northern Norway Lynge B.K. 1,2 Berntsen J. 3 Gjevik B. 1 Department of Mathematics, University of Oslo 1 Norwegian Hydrographic Service 2 Department of Mathematics, University of Bergen 3 JONSMOD 10.-12. May 2010
Outline Introduction-Motivation 1 Introduction-Motivation 2 3 4 5
Outline Introduction-Motivation 1 Introduction-Motivation 2 3 4 5
Introduction - Motivation The objective was to investigate the sensitivity of the horizontal dispersion of particle pairs to the grid size x, y and stratification in a numerical ocean model Accurate model prediction of transports in complex current fields can be of considerable value for several practical purposes - oilspill - sea lice from fishfarming - pollutants Tidal currents dominates the current field in many coastal areas in Norway Tidal effects on dispersion and transports are hence of particular interest
The tidal current Moskstraumen in Lofoten area Our focus: horizontal relative dispersion of particle pairs in tidal currents - on a short time scale (one tidal cycle) - when small scale flow features are important The site for our investigation is the Moskstraumen Maelstrom outside Lofoten on the northern coast of Norway Moskstraumen is known for its strong tidal current (3-5 ms 1 ) and whirlpools
Location of Moskstraumen and the model area 30 69 o N 3000 350 300 250 56 58 60 338 62 62 64 À Ö Ø Ë 30 68 o N 30 67 o N 30 2500 400 1500 200 100 200 VESTFJORDEN Lofoten Bodø Narvik 200 150 100 50 332 326 66 68 64 70 72 62 74 76 Ê 70 78 332 80 82 84 86 Ã Ì ËØ 88 90 Ó Æ ÖÚ 3000 2500 2000 1500 1000 500 300 250 200 150 100 10 o E 12 o E Model area 14 o E 16 o E 18 o E 0 0 50 100 150 200 250 300 350 400 Amplitude and co-tidal lines for M 2 (major semi-diurnal tide) land
Strong tidal current in Moskstraumen SAR image (ERS1-SAT) (Wahl 1995) Semi-major current axis for M 2 -tide)
Strong tidal current in Moskstraumen 16 1 m/s 16 1 m/s 14 14 12 12 10 10 y [km] 8 y [km] 8 6 6 4 4 2 2 0 0 2 4 6 8 10 12 14 16 x [km] 0 0 2 4 6 8 10 12 14 16 x [km]
Outline Introduction-Motivation 1 Introduction-Motivation 2 3 4 5
Bergen Ocean Model Bergen Ocean model (BOM) Three dimensional (x,y,z) σ-coordinate model Include non-linear terms, assume hydrostatic pressure and Boussinesq approximation Mode split used to split the 3-D velocity field into its baroclinic part and its depth integrated part
Bergen Ocean Model Horizontal grid resolution ranging from 50 to 800 meters ( x = y) 10 equidistant σ-layers Boundary conditions: tidal elevation represented by the main diurnal constituent M 2 (from Bjørn Gjeviks tidal model - on 500 meters grid), imposed by FRS Simulations with both homogenous conditions and stratification
Reynolds momentum equations u t + u u + w u z fv = 1 p ρ x + z (A u v z ) + F x, v t + u v + w v z + fu = 1 ρ ρg = p z p y + z (A v v z ) + F y, u = (u, v, w) F x, F y A v g f Velocity field Horizontal eddy viscosity terms (Smagorinsky) Veritical viscosity coeffisient Acceleration of gravity Coriolis parameter
Outline Introduction-Motivation 1 Introduction-Motivation 2 3 4 5
Lagrangian tracers where released in the middle of each horizontal cell at 5 m depth Lagrangian tracers where advected passively with the flow over one M 2 period The relative horizontal dispersion is defined by : r i,j = (x i (t) x j (t)) 2 + (y i (t) y j (t)) 2 r 2 (t) has been calculated for a varying initial distance (δ) between the particles
Mean relativ dispersion in Moskstraumen Mean relative horizontal dispersion is computed by: - R 2 (t) = 1 P i j r i,j 2 (t), - where P is number of particle pairs initially released inside a circle with radius 10 km
Outline Introduction-Motivation 1 Introduction-Motivation 2 3 4 5
Spatial variability of relative dispersion y [km] 25 20 15 10 50 40 30 20 5 10 5 10 15 20 25 x [km] 0 =800 m =50 m Mean relativ dispersion [km 2 ] for each grid cell after one M 2 cycle
Mean relative dispersion in Moskstraumen Mean relative dispersion R 2 (T)( 10 6 [m 2 ]) after one tidal cycle (T ) -calculated for different initial displacement of particles (δ) Larger R 2 (T) for smaller grid size x Relative difference gets smaller for larger δ δ [m] x [m] 50 100 200 400 800 1600 2400 3200 50 12.787 18.915 27.517 38.964 56.257 81.186 104.540 125.955 100 15.580 24.525 37.187 54.683 79.980 98.009 111.842 200 14.755 27.010 44.983 71.437 92.568 107.611 400 11.589 23.118 45.051 65.133 84.470 800 5.689 15.036 26.928 38.455
Mean relative dispersion R 2 (t) in Moskstraumen, fixed δ=800 m 10 7 R 2 [m 2 ] 10 6 0 2 4 6 8 10 12 Time [hours] x=50m x=100m x=200m x=400m x=800m
Lyapunov- and power law exponent, δ = 800m For analysing separation statistics - The Lyapunov exponential model - R 2 L (t) R2 (0)e 2λt - The power law model - R 2 P (t) R2 (0) + t c 1 Estimate the Lyapunov exponent λ and the exponent c 1 The table shows the sensitivity of λ and c 1 to grid size x for δ=800 m x δ λ c 1 strat hom strat hom [m] [m] [day 1 ] [day 1 ] 50 800 4.260 3.932 1.611 1.591 100 800 4.171 4.135 1.606 1.604 200 800 4.151 4.114 1.579 1.571 400 800 3.518 3.895 1.538 1.541 800 800 2.079 2.610 1.415 1.441
The Lyapunov exponent λ as a function of x and δ 8 7 6 x=50 x=100 x=200 x=400 x=800 λ(δ) [day 1 ] 5 4 3 2 1 10 1 10 2 10 3 10 4 δ [m] The rate of growth of mean relative dispersion increases with finer grid resolutions and as δ 0 Convergence of λ for the different x as δ increase
The sensitivity of R 2 (t) to the effects of stratification 10 7 10 7 R 2 [m 2 ] 10 6 R 2 [m 2 ] 10 6 10 5 10 5 stratified x=100 homogeneous x=100 10 4 0 2 4 6 8 10 12 Time [hours] stratified x=800 homogeneous x=800 10 4 0 2 4 6 8 10 12 Time [hours] x = δ = 100 m x = δ = 800 m R 2 (t) (and λ) tend to be somewhat larger under stratified conditions for smaller grid size While for larger x stratification tend to act against dispersion
Density field Introduction-Motivation Density field along a cross-section (RHOSEC) at maximum outflow, for x = 800 m and x = 50 m 0 0 25.7 Depth [m] 50 100 150 26.9 26.8 Depth [m] 50 100 150 26.9 26.8 200 0 5 10 15 20 25 30 Distance [km] 200 0 5 10 15 20 25 30 Distance [km]
Outline Introduction-Motivation 1 Introduction-Motivation 2 3 4 5
Introduction-Motivation The horizontal mean relative dispersion in Moskstraumen, on a time scale of one tidal cycle (T ), is highly dependent on grid resolution x The finer grid resolutions gives the largest R 2 (T) and hence also largest growth of R 2 (t) shown by the Lyapunov exponet λ and the power law exponent c 1 We need to resolve the small scale eddies and complexity of the current field with grid resolution of at least 50-100 m Dispersion is less sensitive to stratification than to grid resolution for the range of grid sizes applied here
Introduction-Motivation Simulations with stratification gives somewhat increased dispersion for the finer grid resolutions The increased dispersion may be explained by increased vertical mixing due to stratification, resolved for the finer grid sizes With higher spatial resolution small scale features, such as internal waves and flow separation, may be represented There are still unresolved processes that may be important for the small-scale mixing and the dispersion of particles In future studies with smaller grid size, it may be necessary to include non-hydostatic pressure effects