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BCCSTECHNICAL REPORT SERIES

Internal waves and internal solitones shoalingand breaking along a continental slope.

Thiem, Ø., Avlesen, H., Alendal, G. andBerntsen, J

REPORT No. 14 15th December 2005

Deliverence to Norsk Hydro.Internal waves.

Contract number 5294251

UNIFOBthe University of Bergen research company

BERGEN, NORWAY

BCCS Technical Report Series is available at http://www.bccs.no/publications/

Requests for paper copies of this report can be sent to:Bergen Center for Computational Science, Høyteknologisenteret,

Thormøhlensgate 55, N-5008 Bergen, Norway

Executive summaryMeasurements in the Ormen Lange area, at the continental shelf slope outside mid Nor-way, have revealed several occurrences of high speeds near the seabed often connected totemperature or salinity variations.

In this report it is investigated if internal waves or internal solitons can lead to themeasured events. Since these kind of waves exists in all oceans and can propagate forhundreds of kilometers without significantly loosing the amplitude the generation area doesnot need to be located close to where the shoaling occurs.

The numerical results show that the shoaling of internal waves or solitons along a shelfslope can lead to breaking and generation of boluses that can propagate up the slope. Duringthese events the maximum horizontal velocity of the wave can be intensified up to almost10 times. The results show that the maximum velocity during a breaking and run up eventdepends on the amplitude of the wave and the steepness of the slope. The combination ofa gentle slope and a internal wave with big amplitude leads to the highest velocities. Thesevelocities is shown that can exceed 1.0 ms−1 when the internal wave or the internal solitonhave an amplitude of 100 m and the slope is 0.05.

1 IntroductionMeasurements in the Ormen Lange area, at the continental shelf slope outside mid Nor-way, have revealed several occurrences of high speeds near the seabed often connected totemperature or salinity variations. In earlier reports [Eliassen and Berntsen, 2000, Eliassenet al., 2000, Eliassen, 2001, Vikebø et al., 2001a,c, Thiem et al., 2001, Sørflaten et al., 2001,Berntsen et al., 2001, Thiem and Berntsen, 2002, Thiem et al., 2002b,a] low pressure ac-tivity which can result in gravity currents [Vikebø et al., 2001b] have been used to explainthese fluctuations in velocity and density.

Some of the events however, is measured when the low pressure activity is of relativeminor strength, and the atmospheric forcing is therefore not believed to be the direct causeto every event [Thiem et al., 2002a]. In these cases traveling internal waves can be theunknown factor which generate the high near seabed velocities. This kind of waves canexist when the density in the interior of the fluid is not constant, which means that theycan be found almost everywhere in the world oceans and they can travel several hundredkilometers with amplitude greater than 100 m.

The generation of internal waves usually starts with a perturbation of the density strat-ification and the internal wave acts much the same as a surface wave. This perturbationcan be due to riverine or glacial intrusions into coastal water, oscillations of oceanic fronts,or abrupt atmospheric or wind variations. But the main generation mechanism for internalwaves is probably by interaction between barotropic tidal flow with topographic featureslike strait sills, continental slopes or sandbanks [Vlasenko et al., 2000].

If the propagation distance of the internal wave is sufficiently large, internal solitarywaves can arise [Vlasenko and Hutter, 2001]. The amplitude of internal solitary wavescan, as for ordinary internal waves, be greater than 100 m with wave lengths as great as20 km. These solitons can propagate without loosing their form significantly for hundredsof kilometers [Global Ocean Associates, 2002]. If their amplitude and typical length scaleof the vertical stratification are of same order, they are called large-amplitude internal soli-tary waves. For this case ordinary Korteweg-de Vries (KdV) theory is not adequate [Brandtet al., 1997, Vlasenko et al., 2000] but the fully Eulerian equations seem to be a good choicefor describing these solitons.

Solitary waves are often studied in a two layer stratification. In tank experiment bothinternal solitons of depression and elevation have been generated and studied [Segur andHammack, 1982, Kao et al., 1985, Helfrich, 1992, Michallet and Barthélemy, 1998, Michal-let and Ivey, 1999, Grue et al., 2000]. However, solitons of elevation [Bogucki et al., 1997]is rarely observed in the ocean, except in areas where the sea floor is shoaling [Klymak andMoum, 2003]. This is probably due to the fact that the density in most of the oceans canbe divided into two different layers where the upper part is typically thinner that the lowerpart. Under these circumstances only solitons or depression can exist [Helfrich, 1992].

When internal solitary waves of depression enters shoaling waters, and propagate up theslope, they will turn into solibores, often referred to as surges or boluses. Observations ofnear seabed solibores over the continental slope in the Faeroe-Shetland Channel [Hosegoodand van Haren, 2004] show same pattern with an abrupt drop in temperature as found inthe measurements at Ormen Lange and the Svinøy section [Skagseth, 2002]. This canindicate that some of the events measured at the Ormen Lange is due to solibores runningup the continental shelf slope, which leads to an increase in the near seabed velocities, andenhanced mixing and resuspension of the sediments along the shelf slope [Michallet andIvey, 1999, Klymak and Moum, 2003, Hosegood et al., 2004].

That internal waves or internal solitary waves can be a problem for the oil industryis known from e.g. the Andaman Sea. In 1975-76 internal wave currents as high as 1.8ms−1 were reported by Exxon in 600 to 1100 m of water in the Andaman Sea offshoreof Thailand. Exxon concluded with that knowledge of internal wave behaviour would benecessary for the design of future deep water offshore production facilities [Osborne andBurch, 1980].

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In Norwegian coastal waters, the most known areas for internal wave generation isEgga, Moskenes, the Vøring plateau and the Norwegian Trench. The internal wave genera-tion in these areas occurs most frequently in the late summer, when the thermal stratificationis most pronounced and the wind conditions are moderate. The Vøring plateau is located at1300 m depth and generates internal waves that propagate in a variety of directions [GlobalOcean Associates, 2002]. In this area the tide is relatively weak [Orvik et al., 2001] andit is most probably other generation mechanisms than the internal tide that produces theinternal waves in this area.

Other areas known for generating internal waves in the vicinity of Norwegian waters isthe East coast of Iceland and the section between Scotland-Faeroe Islands [Global OceanAssociates, 2002]. Since internal waves and solitons are known to propagate hundreds ofkilometers, these areas can not be ruled out as generation areas for internal wave that caninfluence the current regime along the Norwegian shelf.

Internal waves and solitons that shoales along a shelf slope will generate high veloci-ties and resuspension of sediments [Helfrich, 1992, Bogucki et al., 1997, Hosegood et al.,2004]. This is especially pronounced at the depth where the isopycnal meets the shelfslope. In these areas it is believed that the shelf slope is shaped after the critical angle ofthe internal waves. However, other mechanisms as sub sea slides will also influence theslope angle.

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2 The governing equations and numerical model2.1 The governing equationsThe coordinate system for the present studies is (x,z, t) where x is the horizontal coordi-nate, z the vertical coordinate, and t is time. Using the Boussinesq approximation, themomentum equations may be written

∂U∂t +

∂U2∂x +

∂UW∂z − fV = −

1ρ0

∂P∂x +ν

(

∂2U∂x2 +

∂2U∂z2

)

, (1)

∂W∂t +

∂UW∂x +

∂W 2∂z = −

1ρ0

∂P∂z −

gρρ0

+ν(

∂2W∂x2 +

∂2W∂z2

)

. (2)

For an incompressible fluid the equation of continuity is

∂U∂x +

∂W∂z = 0 . (3)

In the present studies a conservation equation for density is applied

∂ρ∂t +

∂Uρ∂x +

∂Wρ∂z = K

(

∂2ρ∂x2 +

∂2ρ∂z2

)

. (4)

In the equations above U(x,z, t), W (x,z, t) are the velocity components in x and z direc-tions, respectively. P(x,z, t) is the pressure, ρ(x,z, t) the density, ρ0 the reference density,and g is the constant of gravity. The eddy viscosity, ν, and the diffusivity, K, are assumedto be constant and equal horizontally and vertically in the present studies.

The pressure is decomposed into pressure due to the surface elevation, η(x, t), internalpressure, and non-hydrostatic pressure, PNH(x,z, t), according to

P(x,z, t) = gρ0η(x, t)+gZ 0

zρ(x,z′, t)dz′ +PNH(x,z, t) . (5)

The equation of continuity is assured by solving an elliptic equation for the non-hydrostaticpressure. The gradient of this pressure is used in the final stages of a time step to adjust thevelocity components to become divergence free.

In the experiments with bottom drag, the bottom friction is specified by

~τx = ρ0CD|Ub|Ub (6)

and the drag coefficient CD is given by

CD =κ2

(ln(zb/z0))2(7)

where z0 is the bottom roughness parameter, zb is the distance of the nearest grid point tothe bottom, and Ub is the horizontal velocity in this grid point. The von Karman constant κis equal to 0.4.

2.2 The BOM modelThe σ-coordinate ocean model is described in Berntsen [2000] and available fromwww.mi.uib.no/BOM/ . The variables are discretized on a C-grid. In the vertical, thestandard σ-transformation, σ = z−ηH+η , where z is the vertical Cartesian coordinate, η thesurface elevation, and H the bottom depth, is applied. For advection of momentum anddensity a TVD-scheme with a superbee limiter described in Yang and Przekwas [1992]is applied in the present studies. The standard second order POM method is applied to

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estimate the internal pressure gradients [Blumberg and Mellor, 1987, Mellor, 1996]. Themodel is mode split with a method similar to the splitting described in Berntsen et al. [1981]and Kowalik and Murty [1993].

Using a splitting technique, the non-hydrostatic pressure may be computed by insertingthe expressions for non-hydrostatic velocity corrections into the equation of continuity [Ka-narska and Maderich, 2003, Heggelund et al., 2004]. Due to the σ-transformation, the term1

ρ0∂PNH

∂σ

(

∂η∂x +σ

∂D∂x

)

appears in Eq. (1), which complicates the computations considerably.An alternative method, that has been adopted in the present study, is to view the non-hydrostatic pressure directly as P(x,σ, t) or a pressure due to convergence or divergence inthe σ-coordinate system [Berntsen and Furnes, 2005]. The non-hydrostatic pressure mayas before be computed by inserting the expressions for non-hydrostatic velocity correctionsinto the equation of continuity. However, with the present approach the elliptic equation forthe non-hydrostatic pressure get the same structure as the pressure equation in z-coordinatemodels, and this simplifies the computations considerably. Using central differences, thepressure equation in 2D only get four off-diagonal elements. In the 3D case, there will besix off-diagonal elements.

The external time steps are performed with a predictor-corrector method where theleapfrog method is used as predictor and the θ-method, where θ is the degree of implic-itness, is used as corrector. For stability θ must be in the range 0.5 ≤ θ ≤ 1, and in thepresent experiments θ = 1.0 (i.e. fully implicit) [Haidvogel and Beckmann, 1999, Casulli,1999]. The internal time steps are performed with a predictor-corrector method where theleapfrog method is used as predictor and the Adams-Moulton 2-step method is used as acorrector, see Heimsund and Berntsen [2004] for analysis of stability of this pair.

The linear system of equations for the non-hydrostatic pressure is solved with the suc-cessive over-relaxation (SOR) method using an optimal value of ω. The iterations are con-tinued until the residual relative error measured in the 2-norm becomes less than 5×10−6.It has been tested that requesting smaller relative errors only gives minor changes of theresults. In the present applications the number of iterations per time step is in the range 10to 100.

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3 Internal wave and solitary wave breakingPropagation, shoaling and breaking of internal waves and internal solitary waves with dif-ferent amplitudes and for different shelf slope geometries have been studied. The simula-tions have been performed with and without the Coriolis effect (rotation). In the simula-tions the length and depth of the shelf is 1.25 km and 300 m respectively, and the lengthof the shelf slope is given in Table 1. The deepest part of the channel is H = 1200 m. Inthe simulations with Coriolis the deepest part of the channel is 100 km long, while in thesimulations without Coriolis the total channel length is 110.25 km. The horizontal grid res-olution is 25 m. In the vertical 100 sigma layers are used. A long channel is used in thesimulations so the solitons and internal waves are able to travel and develop before shoalingalong the shelf slope. The bottom roughness parameter z0 is set equal to 0.0002m.

Initially the water elevation and the velocities are zero. The density distribution inthe Ormen Lange area is dominated by the Atlantic Water on top of the Norwegian SeaWater with a relatively sharp interface in between. The approximate density of the AtlanticWater and the Norwegian Sea Water is ρAW = 1027.50 kgm−3 and ρNSW = 1028.05 kgm−3respectively. The initial density distribution in the model is calculated according to

ρ(x,z) = ρm +∆ρ2

(

1+ tanh[

z− zi −ζ∆h

])

, ∀x,z , (8)

where the mean density ρm = 12 (ρAW + ρNSW ), the density difference ∆ρ = (ρAW − ρNSW ),the interface depth zi = 550.0 m, and the layer thickness ∆h = 25.0 m. This gives a maininterface thickness of approximately 50 m. The interface displacement near the left wall ofthe tank, ζ, is computed from

ζ = 2a0sech2[ x

2L

]

, (9)

where a0 is the amplitude in m, x is the distance from the left wall, and the half width ofthe solitary wave L is computed from

a0L2 =43

(h1h2)2h2−h1

, (10)

where h1 = 550.0 m, h2 = 650.0 m, see as an example Figure 1 (a) [Segur and Hammack,1982, Helfrich, 1992, Bogucki and Garrett, 1993, Bourgault and Kelley, 2003]. This for-mula underestimates the half-lengths by a factor 2 or more [Bourgault and Kelley, 2003].

When the Coriolis (1.3 ·10−4 s−1) is included in the model, the internal solitary wavesthat are generated by this setup are strongly deformed (Figure 1 (c)) and have little incommon with the typical KdV solution (Figure 1 (b)) This is due the geostrophic balanceset up by the model when there are a depression of the interface at the boundary. Theamplitude of the wave is in this case strongly reduced and since the wave does not look likewhat usually is associated with an internal solitary wave, these waves are here referred toas internal waves.

The internal time step is 2.0 s, and there are 30 external time steps per internal step.In the experiments the horizontal and vertical viscosity and diffusivity are constant and

equal to 0.10 m2s−1 which is at the level of diffusivity in the numerical schemes used in themodel.

The total number of experiments performed is 14.

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Simulation Amplitude (m) Slope s Slope length (m)LA050/LA050C 50 0.10 9000LA075/LA075C 75 0.10 9000

LS05/LS05C 100 0.05 18000LS10(LA100)/LS10C(LA100C) 100 0.10 9000

LS15/LS15C 100 0.15 6000LS20/LS20C 100 0.20 4500ES10/ES10C 100 exp 9000

Table 1: The simulations performed in this report are almost twin experiments run withand without Coriolis. The C in the simulation name shows that the simulation is run withCoriolis.

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4 Model resultsVertical snap shots are shown of the simulation results run with and without Coriolis forthe slopes 0.05, 0.10, and 0.15. The snap shots shown are taken every half hour frombefore breaking. Since the distance the wave will travel before breaking is different in thesimulations, the time of the starting snap shot differs from simulation to simulation.

The physics in the different simulations is very similar. They all show breaking whenthe soliton or internal wave shoals on the shelf slope. The results from the simulations withthe most gentle slope (LS05 and LS05C), stand out since there are produced more boluses,also called surges or bores, during the run up phase. So does also the simulation run withthe steepest slope without Coriolis (LS20) since none boluses is produced. The verticalfigures from the simulation with slope 0.10 is commented upon separately in this section,while the figures from the simulations with slope 0.05 and 0.15 are given as a appendix.

4.1 Results solitons (no Coriolis).The number of boluses that are produced in a breaking event when a soliton shoals on ashelf slope was investigated in Helfrich [1992]. He investigated λ ε [0.05,0.20] and showedthat the number of boluses increases when

λ =L

h2√

1+ 1s2(11)

decreases. Here s is the slope, and L is the length of the soliton and h2 is the depth ofthe deepest layer earlier used in Equation 10. The range of λ in this report is outside therange of Helfrich [1992], see Table 2, but when extrapolating the results from Helfrich[1992] it is clear that between zero and two boluses should be expected in the experimentswith the solitons when the slope changes. From the numerical results, it can be seen thatnone boluses is produced when the slope is 0.20 (LS20) and that only one boluses fullydevelops in simulation LS15. For simulation LS10 also a secondary bolus is present after27 hours and 30 minutes, but this secondary bolus is very weak and the extent of the bolusis much smaller then the first. This secondary bolus occur later in the run up process whenthe interface overshoot the equilibrium level. For the experiment with the most gentleslope (LS05), λ = 0.32, there are traces of more boluses being generated. The first bolusdevelop into a double bolus during the propagation up the slope. A secondary bolus isalso developed, but as for simulation LS10, this is weaker and with less extent. Also thebeginning of a third bolus is present after 30 hours, but this is to weak to develop further.

The numerical simulations show that the maximum velocity generated in the shelf slopedue to the soliton is affected by the steepness of the slope, see Figure 2, and by the am-plitude of the incoming soliton, see Figure 3. In the simulation with the most gentle slope(LS05) the deep ocean cross shelf velocity and the vertical velocity are intensified an or-der of 5.5 and 32 times respectively, during the breaking event. The maximum/minimumhorizontal and vertical velocities in the simulations are found in Table 2.

When the soliton enters shallow waters, Figure 2 (a) shows that the velocity of thesoliton is reduced. Figure 2 (a) also shows that parts of the soliton is reflected from the slopeand propagate back into the deep ocean again. The reflection seems more instantaneous forsteep slopes, and a little increase in the cross shelf velocity can be seen when the reflectedsoliton propagate away from the shelf slope, probably due to a thicker interface after thesoliton have passed once. After 47 hours, Figure 2 (a) shows that the soliton is also reflectedfrom the deep ocean boundary. Since the deep ocean length differs in the simulations withdifferent slopes, it can be seen that the soliton is reflected at different times from the deepsea boundary.

The cross shelf velocity of the soliton is constant for the incoming solitons in Fig-ure 2 (a). The vertical velocity however is increased on the solitons way towards the shelf

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slope. This means that the soliton does not change the form to much, but some changes arepresent. The changes consists of a steepening in the rear part of the soliton and a flatteningin the front part.

For the most gentle slope (LS05), Figure 2 shows more fluctuations in the maximumspeed. This means that there are more turbulence and mixing present during the breakingand generation of the boluses on more gentle slopes.

When the amplitude of the incoming soliton increases, the maximum velocity also dras-tically increases (Figure 3). The figure shows that the bigger amplitude, the higher intensi-fication factor should be expected.

4.1.1 Detailed observations of simulation LS10

When the soliton shoals along the shelf slope the front of the interface becomes vertical,Figure 6. The cross shelf velocities are then off shelf above the interface and in front of thevertical interface. The flow of deeper and heavy water is directed toward the shelf. In frontof the vertical interface there are up flow.

As the soliton propagates toward more shallow water a dome in the interface is gen-erated with a vertical front (Figure 7). The vertical interface in front of the dome is nowvery sharp. The negative cross shelf velocity in front of the dome is reduced and the offshelf velocity above the dome increases. Below the interface there is an increase in the onshelf velocity. The water in the front of the dome is moving upward and behind the domedownward.

Figure 8 shows that the effects described in last paragraph increases. The maximumcross shelf velocity is now found in the back part of the dome almost 50 m above the seabed. The high horizontal velocity directed toward the shelf under the dome, has decreased.So have also the vertical upward velocity in front of dome, while the vertical down flowbehind the dome has increased.

After 27 hours, Figure 9 the dome starts propagating upslope, and is now called a bolus,surge, or bore. Since the bore is produced by a soliton, it is also sometimes referred to asa solibore. The maximum horizontal velocity is now found in the back center of the bolusapproximately 50 m above the seabed. The negative horizontal velocity in front of thebolus is now disappeared. The vertical velocity is up in front of the bolus and down behind.Above the bolus it is a negative cross shelf velocity.

A secondary bolus start to develop behind the first bolus, see Figure 10. This is muchweaker and disappears relatively fast. Under this secondary bolus a negative cross shelfflow is seen, transporting water of high density back to its equilibrium level. The maximumhorizontal velocity in the primary bore is now in the center of the bore. Above the bore thehorizontal off shelf flow reduced. So is also the vertical velocities.

Figure 11 and Figure 12 shows that when the bolus looses its momentum the maximumvelocity moves from the center of the bolus toward the seabed. The water with higherdensity then starts to move toward the level of equal density.

4.2 Results internal waves.As for the solitons, the maximum velocity produced by the shoaling of the internal wavesdepends on the amplitude of the incoming internal wave and the shelf slope. This is seenin Figure 4 and Figure 5 and is in good agreement with the results for the solitons. Forthe simulation with the most gentle slope (LS05C) the deep ocean horizontal velocity isintensified an order of 10 times during the breaking event, and since this internal waveproduces two boluses these high velocities occur as two peaks in Figure 4.

The maximum velocities in these simulations is lower than the maximum velocities inthe twin experiments, but this is most probable due to the smaller amplitude of the internalwaves. If the amplitude of the internal wave and the soliton were the same, the maximumvelocity would also probably be approximately the same too. The maximum and minimum

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Simulation max u min u max w min w λLA050 0.1549 -0.1196 0.0897 -0.0439 0.89LA075 0.3536 -0.2022 0.1878 -0.0879 0.76LS05 1.0934 -0.7351 0.6484 -0.6258 0.32

LS10/LA100 0.5571 -0.4240 0.3413 -0.1571 0.63LS15 0.4262 -0.3062 0.2019 -0.0576 0.94LS20 0.3236 -0.2309 0.1331 -0.0435 1.25ES10 0.4417 -0.3034 0.2099 -0.0550 -

Table 2: Maximum and minimum velocities and λ for the different simulations run withoutCoriolis. Cross shelf max/min velocities is marked with u, and vertical max/min velocitiesis marked with w. λ is the nonlinearity parameter defined in Helfrich [1992], see Equa-tion 10.

velocities for the different simulations is found in Table 3, and the table shows that it isalmost a doubling in velocity when the slope is decreased with 0.05. When the initialamplitude of the internal wave is increased with 25 m the velocity is almost doubled in thesimulations.

In Figure 4 it is seen that the internal wave changes the velocity when the wave is inthe deep part of the channel. This indicate that the wave also changes the shape during thepropagation toward the shelf slope.

4.2.1 Detailed observations of simulation LS10C.

Before breaking of the internal wave, high velocities directed toward the deep ocean is seenclose to the shelf slope (Figure 14). This is connected to the out flow of water when thedepression arrives. There is also an in flow along the shelf slope under the interface. Thevertical velocities are small at this stage.

Half an hour later (after 28 hours and 20 min), evidence of overturning is present (Fig-ure 14). A strengthening of the lower part of the interface is seen, and the interface de-velops a vertical front. A dome is developing, and it is an increase in both the horizontaland vertical velocities. The horizontal velocities show that the flow in front of the domeis off shelf close to the slope, and behind the dome the flow is directed on shelf. The onshelf velocities stretches higher up in the water column than the off shelf velocities. Themaximum velocity is still along the seabed, but there is an increase in speed higher up inthe dome. Over the dome there is a weak off shelf velocity, probably to conserve mass.

Figure 15 shows overturning and trapped water in the dome. The horizontal velocitiesshows that the maximum velocity is found under the dome and in the dome. They aredirected on shelf. The off shelf velocities in front of the dome is reduced. There is anincrease in the off shelf velocity above the dome. In the vertical velocities it is evident thatthere is an up flow in the front part of the dome, and a down flow behind the dome.

After 29 hours and 20 minutes (Figure 16) the dome meets the continental shelf slope.The 1027.7 kgm−3 water that was trapped in the dome has been mixed down. The max-imum horizontal velocity is still found in the core of the dome. The off shelf flow alongthe shelf slope in front of the dome is significantly reduced. Over the dome it is a off shelfhorizontal velocity, with vertical flow up in the domes front front and down in the domesback.

In Figure 17, the dome has started moving up the shelf slope. This is often referredto as the head of a bolus, surge or bore. This head now blocks for the off shelf flow thatwas earlier in front of the dome, and there is no off shelf flow in front of the solibore. Themaximum horizontal velocity is found in the core of the bolus. There is vertical transport ofwater up in the bolus front and down in the bolus back. Over the bolus, there is horizontaltransport off shelf.

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Simulation max u min u max w min wLA050C 0.1235 -0.1064 0.0517 -0.0154LA075C 0.2770 -0.2078 0.1529 -0.0685LS05C 0.8366 -0.6969 0.4481 -0.3942

LS10C/LA100C 0.4682 -0.3873 0.2932 -0.1747LS15C 0.2635 -0.2241 0.1585 -0.0558LS20C 0.1676 -0.1881 0.0911 -0.0312ES10C 0.2529 -0.1881 0.1349 -0.0504

Table 3: Maximum and minimum velocities and λ for the different simulations run withCoriolis. Cross shelf max/min velocities is marked with u, and vertical max/min velocitiesis marked with w.

The primary bolus is moving up the slope with the same characteristics as describedin the last paragraph (Figure 18). Behind the head, there is a flow down slope which isdue to the heavier water in the surge flowing down again toward its equilibrium level. Asecondary solibore is also present with maximum horizontal velocity of 0.1 ms−1.

In Figure 19 the head of the bolus is still moving up the shelf slope. The maximumhorizontal velocity is still in the center of the solibore, but is now closer to the shelf slope.The maximum velocity reduces when the bolus propagate upslope. Behind the head, thereis an increasing flow of heavy water down slope.

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5 DiscussionIn the Svinøy section and the Ormen Lange area, data acquisition programs have revealedthat high velocities occur near the seabed [Skagseth, 2002]. The high velocities often oc-cur together with abrupt variations in the temperature and salinity which indicate verticalmovement of the water masses.

Earlier reports have looked to the atmospheric forcing and gravity currents to explainthe high velocities near the seabed in these areas [Eliassen and Berntsen, 2000, Eliassenet al., 2000, Eliassen, 2001, Vikebø et al., 2001a,c, Thiem et al., 2001, Sørflaten et al., 2001,Berntsen et al., 2001, Thiem and Berntsen, 2002, Thiem et al., 2002b,a]. The atmosphericforcing in the Norwegian Sea is a significant force and is probably the cause behind someof the measured events. On the other hand, some of the events have occurred when theatmospheric forcing were relatively weak, which indicate that there must be other unknownmechanisms present to initiate the events.

Propagating density fronts, or internal waves and internal solitons, can have the energyto generate high near seabed velocities and are often measured as rise or drop in tempera-ture and/or salinity [Bogucki et al., 1997, Klymak and Moum, 2003]. Internal waves andinternal solitons exist in all oceans and are measured with amplitudes of more than 100 m[Global Ocean Associates, 2002]. Internal solitons can propagate several hundred kilome-ters, with small changes in the solitons shape. When a soliton meets a continental slope,the soliton will break and solibores or boluses are generated. The number of boluses thatare generated during a breaking event depends on the slope, the depth of the two layers andthe amplitude of the wave [Helfrich, 1992]. The soliton simulations performed in this studyfall outside the range of parameters studied by Helfrich [1992], but it is reason to believethat between zero and two boluses should appear. This is also the case for the soliton simu-lations where the results show that for the steepest slope (0.20) none boluses appears whilefor the most gentle slope (0.05) two boluses appears and a third bolus is under generationbut never develops.

Also internal waves is investigated in this report. It is shown that internal waves alsogenerate boluses during the breaking and run up phase. The simulations show that sig-nificant rise in the velocities appear during breaking of a soliton or an internal wave andthe bore will propagate up the shelf slope. The maximum speed depends on the slope andthe amplitude of the incoming soliton or wave. Two reference runs have been performedwhere the slope is 0.10 and the initial amplitude of the wave is initially 100 m. The differ-ence is that the Coriolis effect has been turned off in one of the simulations. This resultsin that the wave is a soliton, while in the run with Coriolis the internal wave have not theKdV solution shape and is therefore not a soliton. Running the simulation with rotationalso change the amplitude of the wave, which becomes less than in the corresponding sim-ulation without rotation. This can for instance be seen in the maximum velocities. Forthe reference runs the maximum cross shelf velocity are approximately 0.56 ms−1 for thesoliton and 0.47 ms−1 for the internal wave. The maximum vertical velocities are approx-imately 0.34 ms−1 for the soliton and 0.29 ms−1 for the internal wave. The simulationsshow that there will be a significant increase in the velocities if the slope is more gentle orif the amplitude of the incoming soliton or the internal wave is increased.

Maximum velocities above 1.0 ms−1 during the run up phase, is shown to be producedfor a combination of gentle slopes and waves with big amplitudes. These waves haverelative small velocities (maybe 10 times less than during shoaling) and can have lengthsof several kilometers in the deep ocean. They can therefore pass unnoticed on its wayto more shallow areas. To implement a monitoring system for detecting these waves is achallenge.

When waves with wave lengths of several kilometers shoals along the slope, the resultcan be bores which have a size of some hundred meters. Seismic surveys in the Storeggaregion suggest presence of waves and bores of this length scale in the isopycnal [Holbrookand Fer, 2005].

12

This work has been done with the use of a two dimensional slice model. This modelis efficient when looking at different slopes, but excludes the three dimensional effects.When an internal wave or a soliton hits the slope with a horizontal angle, this will influencethe generation of bores and the maximum velocity. Also the effect of the along slopetopography should be an object for closer research. This is specially the case when ainternal wave or soliton is focused in a canyon or in a curvature in the bottom topography.

The generation of internal waves and solitons are often assumed to be connected tointernal tides, but also other mechanisms as sudden atmospheric and wind variations oroscillations of the oceanic fronts can be the generator [Vlasenko et al., 2000]. The lattershould be studied closely since earlier reports [Vikebø et al., 2001c, Thiem et al., 2001,Thiem and Berntsen, 2002, Thiem et al., 2002b,a] have shown that the atmospheric forcingin the Norwegian Sea can generate oscillations of the isopycnal along the shelf slope.

In the area close to Ormen Lange the Vøring plateau is a known source to internalwaves and solitons [Global Ocean Associates, 2002]. However other sources further awayshould also be checked out since they can be able to generate and send internal waves orsolitons in the direction of the Ormen Lange area.

13

0 20 40 60 80 100−1200

−1000

−800

−600

−400

−200

0

Distance [km]

Dept

h [m

]

Time: 00h 00min

(a) Initial stratification.

0 20 40 60 80 100−1200

−1000

−800

−600

−400

−200

0

Distance [km]

Dept

h [m

]

Time: 07h 00min

(b) Stratification after 7 hours run without Coriolis.

0 20 40 60 80 100−1200

−1000

−800

−600

−400

−200

0

Distance [km]

Dept

h [m

]

Time: 07h 00min

(c) Stratification after 7 hours run with Coriolis.

Figure 1: Amplitude 100 m and slope 0.10.

14

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

Time [hours]

Velo

city

[ms−

1 ]

0.050.100.150.20exp

(a) Maximum velocity cross shelf.

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time [hours]

Velo

city

[ms−

1 ]

0.050.100.150.20exp

(b) Maximum vertical velocity.

Figure 2: Maximum velocities for the different slopes used in the simulations with noCoriolis.

15

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

Time [hours]

Velo

city

[ms−

1 ]

100 75 50


0 10 20 30 40 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time [hours]

Velo

city

[ms−

1 ]

100 75 50


Figure 3: Maximum velocities for different amplitudes (m) with no Coriolis.

16

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [hours]

Velo

city

[ms−

1 ]

0.050.100.150.20exp


0 10 20 30 40 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time [hours]

Velo

city

[ms−

1 ]

0.050.100.150.20exp


Figure 4: Maximum velocities for the different slopes for the simulations with Coriolis.

17

0 10 20 30 40 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Time [hours]

Velo

city

[ms−

1 ]

100 75 50


0 10 20 30 40 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time [hours]

Velo

city

[ms−

1 ]

100 75 50


Figure 5: Maximum velocities for different amplitudes (m) for simulations with Coriolis.

18

104 104.2 104.4 104.6 104.8 105−800

−780

−760

−740

−720

−700

−680

−660

−640

−620

−600

Distance [km]

Dept

h [m

]

Time: 25h 30min

1028.05

10281027.95

1027.9 1027.851027.8

1027.75 1027.7

1027.651027.6

1027.55

104 104.2 104.4 104.6 104.8 105−800

−780

−760

−740

−720

−700

−680

−660

−640

−620

−600

Distance [km]

Dept

h [m

]

Time: 25h 30min

−0.1 −0.1−0.05

−0.15

−0.1−0.0

5

0.35

0.3

0.05

00

0.050.10.150.2

0.25

104 104.2 104.4 104.6 104.8 105−800

−780

−760

−740

−720

−700

−680

−660

−640

−620

−600

Distance [km]

Dept

h [m

]

Time: 25h 30min

0

0.05

0.10.15

0.05

Figure 6: Internal soliton (no Coriolis). The slope is 0.10 and the amplitude 100 m. Theupper panel is the density, the mid panel is the cross shelf velocity (ms−1), and the lowerpanel is the vertical velocity (ms−1).

19

104.8 104.9 105 105.1 105.2 105.3 105.4 105.5 105.6 105.7−700

−650

−600

−550

Distance [km]

Dept

h [m

]

Time: 26h 00min

1028

1027.95 1027.9

1027.85 1027

.8102

7.75

1027.7

1027

.65

1027

.6

1027.5

5

104.8 104.9 105 105.1 105.2 105.3 105.4 105.5 105.6 105.7−700

−650

−600

−550

Distance [km]

Dept

h [m

]

Time: 26h 00min

−0.1

−0.05

−0.2−0.15

−0.1−0.05

−0.05

0

0.05

0

00.1

0.5

0.45

0.4

0.450.4

0.40.35

0.3 0.250.2

0.15 0.10.05

0

104.8 104.9 105 105.1 105.2 105.3 105.4 105.5 105.6 105.7−700

−650

−600

−550

Distance [km]

Dept

h [m

]

Time: 26h 00min

−0.05

00.

050.05 0

0.05

0.1

0.15

0.20.25

0.30.3

0.050.1

0.15


20

105.5 105.6 105.7 105.8 105.9 106 106.1 106.2−650

−600

−550

−500

−450

Distance [km]

Dept

h [m

]

Time: 26h 30min

1028.05

10281027.951027

.9 1027.85102

7.81027.7

5

1027.71027.65

1027.61027.55

105.5 105.6 105.7 105.8 105.9 106 106.1 106.2−650

−600

−550

−500

−450

Distance [km]

Dept

h [m

]

Time: 26h 30min

−0.2−0.15

−0.1 −0.0

5

−0.05

0.30.35

0.4

0.450.5

00.0

50.1

0.150.2

0

0.05

0.1

0.15

0.35

0.55

0.35

0.40.450.5

0.4 0.35

105.5 105.6 105.7 105.8 105.9 106 106.1 106.2−650

−600

−550

−500

−450

Distance [km]

Dept

h [m

]

Time: 26h 30min

−0.1 −0.05

00.0

5

0.250.2

0.150.1

0.05

0

0.050.1


21

106 106.1 106.2 106.3 106.4 106.5 106.6 106.7−600

−580

−560

−540

−520

−500

−480

−460

−440

−420

−400

Distance [km]

Dept

h [m

]

Time: 27h 00min

1028.05

1027.75

1027.7

1027.6

5

1027.6

1027.5

5

1027.8

1027.85 102

7.9 1027.95

1028

106 106.1 106.2 106.3 106.4 106.5 106.6 106.7−600

−580

−560

−540

−520

−500

−480

−460

−440

−420

−400

Distance [km]

Dept

h [m

]

Time: 27h 00min

−0.15

−0.15 −0.1−0.05

−0.05

0

0.20.15

0.25

0.30.35

0.4

0.45

0.50.0

5

0.10.1

5

0.2

0

0.05

0.10.150.2

0.250.35

0.3

106 106.1 106.2 106.3 106.4 106.5 106.6 106.7−600

−580

−560

−540

−520

−500

−480

−460

−440

−420

−400

Distance [km]

Dept

h [m

]

Time: 27h 00min

−0.15

−0.1

−0.05

0.15

0.1

0.05

0.05

0


22

106.5 106.6 106.7 106.8 106.9 107 107.1 107.2 107.3

−540

−520

−500

−480

−460

−440

−420

−400

−380

Distance [km]

Dept

h [m

]

Time: 27h 30min

1027

.55

1027

.6

1027.6

5102

7.71027.75

1027.8

1027.95 1027

.9

106.5 106.6 106.7 106.8 106.9 107 107.1 107.2 107.3

−540

−520

−500

−480

−460

−440

−420

−400

−380

Distance [km]

Dept

h [m

]

Time: 27h 30min

−0.15

−0.1−0.05

−0.15

−0.05

−0.10.150.1

0.05

0

0.05

0.1

0.150.250.30.40.3

5

0.3

0.25

0.20.150.10.050

0

106.5 106.6 106.7 106.8 106.9 107 107.1 107.2 107.3

−540

−520

−500

−480

−460

−440

−420

−400

−380

Distance [km]

Dept

h [m

]

Time: 27h 30min

−0.1

−0.05

0

0

0

0.150.1 0.

05


23

107.3 107.4 107.5 107.6 107.7 107.8 107.9

−460

−440

−420

−400

−380

−360

−340

−320

−300

Distance [km]

Dept

h [m

]

Time: 28h 00min

1027.7

1027.7

1027.65

1027

.6

1027.55

107.3 107.4 107.5 107.6 107.7 107.8 107.9

−460

−440

−420

−400

−380

−360

−340

−320

−300

Distance [km]

Dept

h [m

]

Time: 28h 00min

−0.1−0.05

−0.05

0

0.3

0.25

0.2

0.15

0.1

0.05

107.3 107.4 107.5 107.6 107.7 107.8 107.9

−460

−440

−420

−400

−380

−360

−340

−320

−300

Distance [km]

Dept

h [m

]

Time: 28h 00min

−0.05

0.1

0.050


24

107.6 107.7 107.8 107.9 108 108.1 108.2

−420

−400

−380

−360

−340

−320

−300

−280

Distance [km]

Dept

h [m

]

Time: 28h 30min

1027

.651027

.6

1027.5

5

107.6 107.7 107.8 107.9 108 108.1 108.2

−420

−400

−380

−360

−340

−320

−300

−280

Distance [km]

Dept

h [m

]

Time: 28h 30min

−0.05

0

0

0.05

0.10.1

5 0.2

107.6 107.7 107.8 107.9 108 108.1 108.2

−420

−400

−380

−360

−340

−320

−300

−280

Distance [km]

Dept

h [m

]

Time: 28h 30min

−0.05 0.050

Figure 12: Internal soliton (no Coriolis). Slope is 0.10 and amplitude 100 m. Upper panelis the density, mid panel is the cross shelf velocity (ms−1), and lower panel is the verticalvelocity (ms−1).

25

104 104.5 105 105.5 106 106.5−700

−680

−660

−640

−620

−600

−580

Distance [km]

Dept

h [m

]

Time: 27h 50min

1028

1027.951027.9

1027.851027.8

1027.75

1027.7

1027.65

1027.6

1027.55

104 104.5 105 105.5 106 106.5−700

−680

−660

−640

−620

−600

−580

Distance [km]

Dept

h [m

]

Time: 27h 50min

−0.05

−0.1

−0.3

−0.1

5 −0.2 −

0.25

0

0.05 0

0.05

0.1

0.15

104 104.5 105 105.5 106 106.5−700

−680

−660

−640

−620

−600

−580

Distance [km]

Dept

h [m

]

Time: 27h 50min

0

0.05

0

Figure 13: Internal wave (with Coriolis). The slope is 0.10 and the amplitude 100 m. Theupper panel is the density, the mid panel is the cross shelf velocity (ms−1), and the lowerpanel is the vertical velocity (ms−1).

26

105 105.5 106 106.5−660

−640

−620

−600

−580

−560

−540

−520

Distance [km]

Dept

h [m

]

Time: 28h 20min

−0.05

−0.2

−0.15

−0.1−0.

05

0

−0.05−0.1

0.350.3

5

0.30.25

0.20.150.1

0.05

0.05

0.15

0.050.

1

0

105 105.5 106 106.5−660

−640

−620

−600

−580

−560

−540

−520

Distance [km]

Dept

h [m

]

Time: 28h 20min

−0.0

5

0

0.05

0.10.15

0.2

0.25

0.05

0

0

0.1


27

105.5 106 106.5−620

−600

−580

−560

−540

−520

−500

−480

Distance [km]

Dept

h [m

]

Time: 28h 50min

10281027.951027

.91027.851027.81027.75

1027.651027.7

1027.6

1027.55

1027

.7

1027

.65

1027.7

105.5 106 106.5−620

−600

−580

−560

−540

−520

−500

−480

Distance [km]

Dept

h [m

]

Time: 28h 50min

−0.1−0.0

5

−0.05−0.1

−0.15−0.2

−0.05−0.05

0

00.05

0.1

0.25

0.30.3

5

0.4

0.250.3

0.35

0.4

0.3

0.250.2

0.15

0.1

0.05

0

0.15

0.25

105.5 106 106.5−620

−600

−580

−560

−540

−520

−500

−480

Distance [km]

Dept

h [m

]

Time: 28h 50min

−0.1

5−0

.1

−0.05

0

0

0

00.

050.10.150.2


28

106 106.5 107 107.5−580

−560

−540

−520

−500

−480

−460

−440

Distance [km]

Dept

h [m

]

Time: 29h 20min

1027.7

1027.55 1

027.

610

27.6

5

10281027.95102

7.91027.851027.81027.751027.7

1027.65

106 106.5 107 107.5−580

−560

−540

−520

−500

−480

−460

−440

Distance [km]

Dept

h [m

]

Time: 29h 20min

−0.05

−0.15

−0.1−0.05

0

0.2

0.25

0.35

0.4

0.25

0.3

0.25

0 0.05

0.10.15

0.05

0.10.15

0.2

106 106.5 107 107.5−580

−560

−540

−520

−500

−480

−460

−440

Distance [km]

Dept

h [m

]

Time: 29h 20min

−0.1

−0.0

5

0

0.05

0

0.150.1

0.05


29

106.5 107 107.5−520

−500

−480

−460

−440

−420

−400

Distance [km]

Dept

h [m

]

Time: 29h 50min

10281027.95

1027.91027.851027.81027.751027.7

1027.651027.6

1027.55

1027.651027.6

1027.55

1027.7

106.5 107 107.5−520

−500

−480

−460

−440

−420

−400

Distance [km]

Dept

h [m

]

Time: 29h 50min

−0.15−0.1

−0.0

5 0

0.05

0

0.15

0.15

0.1

0.05

0

0.10.0

5

0.20.25

0.3

0.35

0.3

0

0.10.150.2

0.25

106.5 107 107.5−520

−500

−480

−460

−440

−420

−400

Distance [km]

Dept

h [m

]

Time: 29h 50min

−0.1

−0.0

5

0

0

0

0.15 0.1

0.05


30

107 107.2 107.4 107.6 107.8 108−500

−480

−460

−440

−420

−400

−380

−360

Distance [km]

Dept

h [m

]

Time: 30h 20min

1028

1027.91027.8

1027.551027.6

1027.65

1027.7

107 107.2 107.4 107.6 107.8 108−500

−480

−460

−440

−420

−400

−380

−360

Distance [km]

Dept

h [m

]

Time: 30h 20min

−0.1−0.05

−0.05

0

0

0

0.05

0.1

0.15

00.05

0.1

0.3

0.25

0.20.15

0.1

0.05

0.10.150.2

0.25

107 107.2 107.4 107.6 107.8 108−500

−480

−460

−440

−420

−400

−380

−360

Distance [km]

Dept

h [m

]

Time: 30h 20min

−0.1

−0.0

5

0

0

00.1

0.05

00


31

107.5 108 108.5

−440

−420

−400

−380

−360

−340

−320

Distance [km]

Dept

h [m

]

Time: 30h 50min

1027

.55

1027

.6

1027

.65

107.5 108 108.5

−440

−420

−400

−380

−360

−340

−320

Distance [km]

Dept

h [m

]

Time: 30h 50min

−0.05

−0.1−0.05

0

0

0.2

0.15

0.1

0.05

107.5 108 108.5

−440

−420

−400

−380

−360

−340

−320

Distance [km]

Dept

h [m

]

Time: 30h 50min

−0.0

5 0

0.05


32

97.5 98 98.5 99 99.5 100 100.5 101 101.5

−850

−800

−750

−700

−650

−600

−550

Distance [km]

Dept

h [m

]

Time: 24h 00min

1028.05

10281027.95

1027.55

1027.61027.65

1027.71027.75

1027.81027.85

1027.9

97.5 98 98.5 99 99.5 100 100.5 101 101.5

−850

−800

−750

−700

−650

−600

−550

Distance [km]

Dept

h [m

]

Time: 24h 00min

−0.5

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

−0.6

0

0 0.40.3

0.20.1

97.5 98 98.5 99 99.5 100 100.5 101 101.5

−850

−800

−750

−700

−650

−600

−550

Distance [km]

Dept

h [m

]

Time: 24h 00min

−0.0

5

0

0

0

0.10.05

0.05

0

00.10


33

97.5 98 98.5 99 99.5 100

−850

−800

−750

−700

−650

−600

−550

Distance [km]

Dept

h [m

]

Time: 24h 30min

1027.55

1027.6

1027.651027.7

1028.05

1028

1027.95

1027.851027.751027.65

1027.55

1027

.75

97.5 98 98.5 99 99.5 100

−850

−800

−750

−700

−650

−600

−550

Distance [km]

Dept

h [m

]

Time: 24h 30min

−0.5−0.4

−0.3−0.2

−0.1

−0.2

−0.3

−0.2

−0.1

−0.2

−0.1

−0.1

−0.1

−0.1

0.2

0.1

0

0

0.2

0.1

0

00.10

0.1

0.2

0.30.40.5

0.6

0.8

0.7

0.80.7

97.5 98 98.5 99 99.5 100

−850

−800

−750

−700

−650

−600

−550

Distance [km]

Dept

h [m

]

Time: 24h 30min

−0.2

−0.1−0.1

0

0

0.1

0.1

0

0

0.20.1

0.1

0

0.5

0.1

0.10.2

0.4


34

98 98.5 99 99.5 100 100.5−850

−800

−750

−700

−650

−600

−550

−500

Distance [km]

Dept

h [m

]

Time: 25h 00min

1027.651027.6

1027.55

1027

.95

1028.051028.05

1027.751027.81027.851027.95

1028

1027.7

5

1027

.65

98 98.5 99 99.5 100 100.5−850

−800

−750

−700

−650

−600

−550

−500

Distance [km]

Dept

h [m

]

Time: 25h 00min

−0.4−0.3

−0.2−0.1

−0.1−0.1

−0.1

−0.2−0.1

−0.3

−0.4−0.2−0.1

−0.3

0.1

0.20.3

0.3

0.80.8 0.6

0.6

0.6

0.7

0.7

0.80.7

0.60.5

0.40.30.2

0.10

0.20.1

0

0.50.40

.30.2

0.1

0

0 0.1

0

98 98.5 99 99.5 100 100.5−850

−800

−750

−700

−650

−600

−550

−500

Distance [km]

Dept

h [m

]

Time: 25h 00min

−0.4

−0.3

−0.2

−0.1

−0.1

−0.1

−0.1

−0.1

0

0

0.1

0

0.10.1

0 0.1 0

0

0.1

0.2

0.3

0.50.2

0.30.6


35

98.5 99 99.5 100 100.5 101

−800

−750

−700

−650

−600

−550

−500

−450

Distance [km]

Dept

h [m

]

Time: 25h 30min

1027

.95

1027

.8

1027.55

1028.05

1028

1027.95

1027

.5510

27.6

1027.65

1027.95 1028

1027.75

98.5 99 99.5 100 100.5 101

−800

−750

−700

−650

−600

−550

−500

−450

Distance [km]

Dept

h [m

]

Time: 25h 30min

−0.6

−0.1

−0.1−0.1

−0.2−0.1

−0.5

−0.4

−0.2

−0.3

−0.1

0.8

0.7

0.70.6

0.5

0.40.50.6

0.4

0.2 0.1

0

0.3

0.1

0

0.10.2

0.30.40.5

0

0.10.

2

0.3

98.5 99 99.5 100 100.5 101

−800

−750

−700

−650

−600

−550

−500

−450

Distance [km]

Dept

h [m

]

Time: 25h 30min

−0.5

−0.4

−0.3−0.2

−0.10 0

0

0

0

0.10 0.1

0

0

0.1

0.20.3

0.40.5

0

00.1

0.2

0.3


36

99.5 100 100.5 101 101.5 102

−750

−700

−650

−600

−550

−500

−450

−400

Distance [km]

Dept

h [m

]

Time: 26h 00min

1027.55

1027

.95 1027.75

1028.05

1027.55

1027.95102

8

1027.75

99.5 100 100.5 101 101.5 102

−750

−700

−650

−600

−550

−500

−450

−400

Distance [km]

Dept

h [m

]

Time: 26h 00min

−0.6

−0.2 −0.1

−0.1

−0.2

−0.1

−0.2

−0.2−0.1

−0.1

−0.2−

0.3

−0.5−0.3

0.1

0.5

0.4 0.3

0.30.4

0.5

0.80.7

0.6

0.1

0

0.2

0.3

0.50.6

0

0.1

0.2

0.40.30.2

0.1

0

99.5 100 100.5 101 101.5 102

−750

−700

−650

−600

−550

−500

−450

−400

Distance [km]

Dept

h [m

]

Time: 26h 00min

−0.5

−0.4

−0.3−0.2

−0.1

−0.1

0

0.1

0

0 0

0

0

0.1

0.2

0.3

0.40.4

0.50.4

0.20.2

0.1

0


37

100.5 101 101.5 102 102.5 103

−700

−650

−600

−550

−500

−450

−400

−350

Distance [km]

Dept

h [m

]

Time: 26h 30min

1027

.85

1027

.8

1028.05

1027.55

10281027.95

1027.95

1027

.75

100.5 101 101.5 102 102.5 103

−700

−650

−600

−550

−500

−450

−400

−350

Distance [km]

Dept

h [m

]

Time: 26h 30min

−0.5

−0.4 −0.3−0.2

−0.1

−0.1

0

0.4 0.40.

4

0.20.3

0.20.5

0.9

0.8

0.70.6

0

0.1

0.2

0.2

0.3

0.40.5

0.1

0

0.20.3

0.4

0.5

100.5 101 101.5 102 102.5 103

−700

−650

−600

−550

−500

−450

−400

−350

Distance [km]

Dept

h [m

]

Time: 26h 30min

−0.3

−0.3

−0.2

−0.1

−0.3

−0.1

0

0

0 0

0.30.2

0.1 00

0.10.2

0.30.4

0.50.5


38

101.5 102 102.5 103 103.5 104

−650

−600

−550

−500

−450

−400

−350

−300

Distance [km]

Dept

h [m

]

Time: 27h 00min

1027

.85

1027

.8

1027.55

1028

1027

.610

27.65

1027

.75

101.5 102 102.5 103 103.5 104

−650

−600

−550

−500

−450

−400

−350

−300

Distance [km]

Dept

h [m

]

Time: 27h 00min

−0.6−0.5−

0.4

−0.4

−0.1−0.1

−0.3−0.2−0

.1

0.8

0.8

0

0

0.4

0.30.2

0

0.1

0.2

0.3

0.2

0.5

0.3

0.1

0.20.30.4

0.50.

60.7

0

0.1

0.2

0.30.

40.5

0.6

0.7

101.5 102 102.5 103 103.5 104

−650

−600

−550

−500

−450

−400

−350

−300

Distance [km]

Dept

h [m

]

Time: 27h 00min

−0.2

−0.1

−0.1

−0.4

−0.3

−0.2

−0.1

0

0

0

0

0

0

0.1

0.20

0.5

0.4 0.3 0.2

0.1

0

0.3


39

102.5 103 103.5 104 104.5

−600

−550

−500

−450

−400

−350

−300

−250

Distance [km]

Dept

h [m

]

Time: 27h 30min

1027

.65

1027.7

1027

.75

1027.71027.55

10281027.8

102.5 103 103.5 104 104.5

−600

−550

−500

−450

−400

−350

−300

−250

Distance [km]

Dept

h [m

]

Time: 27h 30min

−0.5

−0.4

−0.3

−0.2

−0.1

−0.1

−0.1

−0.1−0.20.2

0.2

00.1

0.1

0.4

0.3

0.9

0.2

0.3

0.4

0.5

0.1

0

0

0.10.20.3

0.3

0.1

0.2

0.3

0.4

0.5

0.80.7

0.6

0.5

0.3

102.5 103 103.5 104 104.5

−600

−550

−500

−450

−400

−350

−300

−250

Distance [km]

Dept

h [m

]

Time: 27h 30min

−0.1

−0.3−0

.2−0

.2

−0.3−0.2−0.1

−0.1

0

0.30.2

0.10

0.2

0.3

0.3

0.2

0.1

0.3

0

0


40

103 103.5 104 104.5 105−600

−550

−500

−450

−400

−350

−300

−250

Distance [km]

Dept

h [m

]

Time: 28h 00min

1027

.65

1027

.65

1027

.7

1027

.65

1027

.6

1027

.7

1027.7

1027.8 1027.55

1027.55

1028

103 103.5 104 104.5 105−600

−550

−500

−450

−400

−350

−300

−250

Distance [km]

Dept

h [m

]

Time: 28h 00min

−0.4−0.3

−0.3−0.2

−0.2−0

.1 −0.1

−0.2

−0.1

−0.10

0.1

0

0.4

0.3

0

0.10.

2

0.1 0

.7 0.7

0.6

0.2

0.30.4

0.5

0.1

0

0.5

0.40.30.2

0.10

0.2

0.3

0.4

0.20.4

103 103.5 104 104.5 105−600

−550

−500

−450

−400

−350

−300

−250

Distance [km]

Dept

h [m

]

Time: 28h 00min

−0.1

−0.4

−0.2

−0.2

−0.1

−0.3

−0.2−0

.1

−0.1

0

0.20.2

0.1

0

0

0.2

0.2

0.1

0

0

0.1

0.2

0.3

0.2

0.1

0

0.2

0.1


41

103.5 104 104.5 105 105.5

−550

−500

−450

−400

−350

−300

−250

Distance [km]

Dept

h [m

]

Time: 28h 30min

1027.55

1027

.610

27.6

1027.6

1027.65

1027

.7

1027.65

1027

.55

1027

.75

103.5 104 104.5 105 105.5

−550

−500

−450

−400

−350

−300

−250

Distance [km]

Dept

h [m

]

Time: 28h 30min

−0.3

−0.3

−0.2 −0.2−0.1

−0.1

−0.2−0.1

−0.1

0.7 0.6

0.6

0.5

0.40.3

0.20.1

0

0.1

0.4

0.2

0.3

0.20.1

0.1

0 00

0.7

0.5 0.2

0.30.4

0.10

0.10.2

0.30.4

0.5

0.6

103.5 104 104.5 105 105.5

−550

−500

−450

−400

−350

−300

−250

Distance [km]

Dept

h [m

]

Time: 28h 30min

−0.3

−0.2

−0.2

−0.1

−0.1

−0.1

0

0.20.1

0

0.1

0

0.2

0.1

0

0

0.2

0.10.2

0.1

0.1


42

104 104.5 105 105.5 106 106.5−550

−500

−450

−400

−350

−300

−250

−200

Distance [km]

Dept

h [m

]

Time: 29h 00min

1027.65

1027.6

1027.55

1027.75

1027.55

1027.7

104 104.5 105 105.5 106 106.5−550

−500

−450

−400

−350

−300

−250

−200

Distance [km]

Dept

h [m

]

Time: 29h 00min

−0.2−0.1

−0.1−0.1

−0.1

−0.1

−0.1

−0.3

−0.3

−0.2

−0.1

0.6

0.6

0

0

0.3

0

0.1

0

0.50.40.3

0.20.1

0

0.1

0.4

0.20.3

0.5

0.5

0

104 104.5 105 105.5 106 106.5−550

−500

−450

−400

−350

−300

−250

−200

Distance [km]

Dept

h [m

]

Time: 29h 00min

−0.2

−0.1

−0.2

−0.1

−0.1

−0.2

0.10.20

0.1

0.1

00

00.20.1

0

0.10

0

0


43

104 104.5 105 105.5 106 106.5 107−550

−500

−450

−400

−350

−300

−250

−200

Distance [km]

Dept

h [m

]

Time: 29h 30min

1027.55

1027.6

5

1027.6

1027.7

1028

1027.55

104 104.5 105 105.5 106 106.5 107−550

−500

−450

−400

−350

−300

−250

−200

Distance [km]

Dept

h [m

]

Time: 29h 30min

−0.3

−0.2

−0.1

−0.2

−0.1

−0.1

−0.1

−0.2−0.1

−0.1

0

0.7

0.1

0

0

0

0

0.1

0

0

0.2

0.1 0 0.1 0

0.2

0.30.

4

0.50.6

0.4

0.3

0.20.1

0.4

0.40.3

0.20.1

0

104 104.5 105 105.5 106 106.5 107−550

−500

−450

−400

−350

−300

−250

−200

Distance [km]

Dept

h [m

]

Time: 29h 30min

−0.1

−0.2

−0.1

−0.2

−0.1

0.10

0

0

0.20.1 0

00

0

0

0.1

0.1

0.20.1

0


44

104 104.5 105 105.5 106 106.5 107 107.5−550

−500

−450

−400

−350

−300

−250

−200

Distance [km]

Dept

h [m

]

Time: 30h 00min

1027.7

1027.55

1027

.55 1027.55

1027.6

1027.65

1027.55

104 104.5 105 105.5 106 106.5 107 107.5−550

−500

−450

−400

−350

−300

−250

−200

Distance [km]

Dept

h [m

]

Time: 30h 00min

−0.3

−0.2

−0.1 −0.1

−0.2

−0.1

−0.1

−0.2

−0.1

−0.2−0.1

−0.1

−0.1

0.2

0.1

0

0

0

0

0

0

0.4

0.5

0.4

0.3

0.2

0

0.1

0.1

0.2 0.3

0.3

0.2

0.1

0

104 104.5 105 105.5 106 106.5 107 107.5−550

−500

−450

−400

−350

−300

−250

−200

Distance [km]

Dept

h [m

]

Time: 30h 00min

−0.2

−0.2

−0.1

−0.1

−0.1

−0.1

−0.1

0

0

0

0.1 00 0

0.2

0.1

00.

2

0.1

0.1

00


45

105 105.5 106 106.5 107 107.5 108−500

−450

−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 30h 30min

1027.55

1027.55

1027.55

1027.6

1027.65

1027.65

1027

.6

105 105.5 106 106.5 107 107.5 108−500

−450

−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 30h 30min

−0.2

−0.2−0.2

−0.1

−0.1

−0.1

−0.3

−0.2

−0.1

−0.1

−0.2−0.1

−0.3

−0.2

−0.10

0.2

0

0.1

0 0

0.3

0.2

0.1

0.4

0.3

0.4

0.3

0

0

0.3

0

0.1 0.2

0.3

0.4

0

0.1

0.2

0.1

0.2

0.2

0.1

105 105.5 106 106.5 107 107.5 108−500

−450

−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 30h 30min

−0.1

−0.2

−0.2

−0.2

−0.2

−0.1 −0

.1

−0.1

−0.1

0.10

0.2

0.1

00.

2

0.100

0

0

0

0

0

00

0.1

0.1

0.1


46

105.5 106 106.5 107 107.5 108 108.5

−450

−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 31h 00min

1027

.55

1027.55

1027.55

1027.6

1027.5510

27.6

1027.55

1027

.61027.65

105.5 106 106.5 107 107.5 108 108.5

−450

−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 31h 00min

−0.1

−0.1−0

.1

−0.2

−0.2

−0.1

−0.1

−0.3

−0.2

−0.1

0.10

0

0

0.3

0.3

0.2

0.3

0.3

0.2

0

0.1

0

00.

1

00.

1

0.1

0

0.10.

2

105.5 106 106.5 107 107.5 108 108.5

−450

−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 31h 00min

−0.1

−0.1

−0.1

−0.1

0.1

0 0

0

0

0

0.1

0.1

0

0

0

0.1

0

00.

1

00.1

0.1


47

105.5 106 106.5 107 107.5 108 108.5 109

−450

−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 31h 30min

1027

.55

1027.55

1027.6

1027

.55

1027

.55

1027.55

1027.55

1027.6

105.5 106 106.5 107 107.5 108 108.5 109

−450

−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 31h 30min

−0.1

−0.1

−0.1

−0.1

−0.1

−0.1

−0.2

−0.2

0.1 0

0

0

0.2

0.3

0.3

0

0.1

0

0

0.20.

10.

1

0.1

0

0.2

0.20.

1

0.1

105.5 106 106.5 107 107.5 108 108.5 109

−450

−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 31h 30min

−0.1

−0.1

−0.1

−0.1

−0.1

−0.1

0

0 0

0

00

00.

1

0

0

00.10.

1

0.1

0

0.1

0

0

0


48

107 107.5 108 108.5 109 109.5−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 32h 00min

1027.55

1027.55 1027

.55

1027

.55

107 107.5 108 108.5 109 109.5−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 32h 00min

−0.1

−0.1−0.1

−0.1−0.1

−0.1

0.3

0.2

0.10

0.20

0.1

0

0.2

0.20.1 0

.1

0.1

0

107 107.5 108 108.5 109 109.5−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 32h 00min

−0.1

−0.1

0

0

0

00

0

0

0

0.1 00.1

0

00

0


49

107 107.5 108 108.5 109 109.5−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 32h 30min

1027

.55

1027.55

1027.55

107 107.5 108 108.5 109 109.5−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 32h 30min

−0.2 −0.2

−0.1

−0.1

−0.1

−0.1 −0.1

0.3

0.2

0.1

0.10.1

0

0

0.1

0.2

0.2

00.

1

0

0.1

0

107 107.5 108 108.5 109 109.5−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 32h 30min

−0.1

−0.1

0

0

0000

0.1

0

0

0

0

0


50

107 107.5 108 108.5 109 109.5−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 33h 00min

1027.55

1027

.55

107 107.5 108 108.5 109 109.5−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 33h 00min

−0.1

−0.1

0

0.1

0.2

0.3

0.2

0.1

0.1

0.2

0.1

0

0.1

0

0

0.1

107 107.5 108 108.5 109 109.5−400

−350

−300

−250

−200

−150

Distance [km]

Dept

h [m

]

Time: 33h 00min

−0.1

−0.1

−0.1

0

00

0 0

0

0

0


51

108.5 109 109.5 110 110.5 111−720

−700

−680

−660

−640

−620

−600

−580

Distance [km]

Dept

h [m

]

Time: 29h 30min

1027.55

1027.6

1027.65

1027.7

1027.75

1027.8

1027.85 1027.9

1027.95

1028

1028.05

108.5 109 109.5 110 110.5 111−720

−700

−680

−660

−640

−620

−600

−580

Distance [km]

Dept

h [m

]

Time: 29h 30min

−0.65

−0.6

−0.5

5−0.5

−0.45

−0.4

−0.3

5−0

.3−0

.25

−0.2

−0.1

5−0

.1

−0.0

5

0

0.05

0.15

0.10.0

5

108.5 109 109.5 110 110.5 111−720

−700

−680

−660

−640

−620

−600

−580

Distance [km]

Dept

h [m

]

Time: 29h 30min

0

0

Figure 39: Internal wave (with Coriolis). The upper panel is the density, the mid panel isthe cross shelf velocity (ms−1), and the lower panel is the vertical velocity (ms−1).

52

109.5 110 110.5 111

−680

−660

−640

−620

−600

−580

−560

−540

−520

−500

Distance [km]

Dept

h [m

]

Time: 30h 00min

10281027.95

1027.9

1027.851027.8

1027.75 1027.7

1027.651027.6

1027.55

109.5 110 110.5 111

−680

−660

−640

−620

−600

−580

−560

−540

−520

−500

Distance [km]

Dept

h [m

]

Time: 30h 00min

−0.05

−0.1

−0.05

−0.05

−0.4

−0.3

5 −0.45

−0.3−0.25−0.2

−0.15−0.1−0.

05

0

0.05

0.10.1

5

0.30.25

0.2

0.35 0.30.25

0.2

0.15 0.1

0.05

00.15 0.1

0.05

0

0.05

0.55

0.2

0.25 0.30.35

0.4

0.45

0.75

00.050.1

0.15

0.2

0.25 0.3

0.4

0.50.6

109.5 110 110.5 111

−680

−660

−640

−620

−600

−580

−560

−540

−520

−500

Distance [km]

Dept

h [m

]

Time: 30h 00min

−0.0

5−0.25

−0.2

−0.1

5

−0.1−0.

05

0

0

0

0

0.15

0.1

0.05

0

0.35

0.1

0.15

0.25

0.35

0.1

0.15

0.05

0


53

110.6 110.8 111 111.2 111.4 111.6−650

−600

−550

−500

−450

Distance [km]

Dept

h [m

]

Time: 30h 30min

1028

1027.95

1027.9

1027.85 1027.8

1027.75

1027.710

27.610

27.55

1027.65

1027.85

1027.551027.6 1027.65

110.6 110.8 111 111.2 111.4 111.6−650

−600

−550

−500

−450

Distance [km]

Dept

h [m

]

Time: 30h 30min

−0.3−0.25

−0.2−0.15

−0.1−0.05

−0.1

−0.05

−0.25−0.2

−0.15

−0.1−0.05

0.55

0.50.45

0.40.

45

0.350.3

0.250.2

0.150.1

0.050

0

0.05

0.1

0.150.2

0.25

0.350.60.5

5

0.50.45

0.40.35

0.3

0.55

0.5

0.45

0.450.4

0.350.3

0.250.2

00.05

0.10.15

0.2

0.35

0.4

110.6 110.8 111 111.2 111.4 111.6−650

−600

−550

−500

−450

Distance [km]

Dept

h [m

]

Time: 30h 30min

−0.2−0.15

−0.1−0.05

−0.0

5

0

0

0.05 0.10.05

0.4

0.35

0.3

0.25

0

0.05

0.1

0.15

0.20


54

111.4 111.6 111.8 112 112.2 112.4

−600

−550

−500

−450

−400

Distance [km]

Dept

h [m

]

Time: 31h 00min

1027.751027.75

1027.551027.6

10281027.95

1027.9

1027.85

1027.551027.6

1027.65 1027.7

1027.651027.7

1027.8

1027.75

111.4 111.6 111.8 112 112.2 112.4

−600

−550

−500

−450

−400

Distance [km]

Dept

h [m

]

Time: 31h 00min

−0.35−0.3−0.25 −0.2 −0.15−0.1 −0.05−0

.05

−0.2−0.15−0.1−0.05

0

0.05

0.1

0.15

0.20.25

0.3

0.35

0.6 0.5

0.550.50.450.4

0.35

0.350.5

0.4 0.45

0.5

0.450.4

0.350.3

0.250.20.150.1

0.05

0

0.40.35

0.30.25

0.65

0.6

0.55 0

.50.

450.4

0

111.4 111.6 111.8 112 112.2 112.4

−600

−550

−500

−450

−400

Distance [km]

Dept

h [m

]

Time: 31h 00min

−0.2

−0.1

5−0

.1−0

.05

−0.2

5−0

.2−0

.15

0

0

0.15

0.1

0.05

0.40.35

0.3 0.250.2

0.15

0.1

0.05

0


55

112 112.2 112.4 112.6 112.8 113 113.2−600

−550

−500

−450

−400

−350

Distance [km]

Dept

h [m

]

Time: 31h 30min

1027.7

1027.7

1027.65

1027.551027.6

10281027.95

1027.751027.81027.9

112 112.2 112.4 112.6 112.8 113 113.2−600

−550

−500

−450

−400

−350

Distance [km]

Dept

h [m

]

Time: 31h 30min

−0.4−0.35−0.3−0.25

−0.2−0.15−0.1−0.05

−0.1−0.050

0

0.05

0.1

0.15

0.2

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