eivind støylen and jan erik h. weber- mean mass transport induced by internal kelvin waves, with...

8/3/2019 Eivind Støylen and Jan Erik H. Weber- Mean mass transport induced by internal Kelvin waves, with application to th…

http://slidepdf.com/reader/full/eivind-stoylen-and-jan-erik-h-weber-mean-mass-transport-induced-by-internal 1/4

1

Mean mass transport induced by internal Kelvin waves,

with application to the circulation in the Van Mijen fjord in Svalbard

Eivind Støylen and Jan Erik H. Weber

Department of Geosciences, Section for Meteorology and Oceanography,

University of Oslo, Norway

Abstract

Waves propagating in a fluid induce a mean drift in the direction of the wave

propagation (Stokes 1847). Stokes considered wave motion in inviscid fluids. Longuet-Higgins

(1953) was the first to demonstrate that the effect of viscosity introduces a mean Eulerian

drift in addition to the Stokes drift in water waves. The present study is concerned with the

mean drift induced by internal Kelvin waves. It appears that the drift due to wind-induced

internal Kelvin waves can explain observed features of the circulation in large lakes, e.g.

Csanady (1972), Wunsch (1973), Ou and Bennett (1979).

In this study we consider internal Kelvin waves which are generated by barotropic

tidal motion over a topographic feature. The stratification is modeled as a two-layer system.

We intend to apply the theory to an Arctic fjord, with a shallow sill at the entrance. The

density difference is such that the internal Rossby radius of deformation is much smaller

than the width of the fjord. In Arctic regions most fjords are ice-covered for a long period of

time. In this period the water mass is sheltered from the wind, and the only driving

mechanism for internal Kelvin waves is the barotropic tide. The generated internal Kelvin

waves have constant frequency, but become spatially damped due to frictional effects. The

main contribution to the damping comes from the stationary ice cover when the upper layer

is shallow. The mean drift in these spatially damped waves may induce a mean circulation in

the fjord basin inside the sill.

As far as the analytical analysis is concerned, the model is a semi-infinite fluid in the

northern hemisphere bounded by a straight coast with the x -axis along the coast, and the y -

axis directed off-shore. The fluid has two layers of constant densities ρ1 and ρ2. The lower

layer is very deep compared to the upper layer thickness H1. The surface is ice-covered, and

modeled as a stationary horizontal rigid lid. The wave motion takes place at the interface ξ

between the layers.



2

We intend to solve the nonlinear equations of motion within an Eulerian framework.

To obtain the mean Lagrangian upper-layer volume fluxes U1 and V 1 due to the wave motion,

we integrate the governing Eulerian equations in the vertical between the material interface

ξ and the surface; see Phillips (1977), or Weber et al. (2006) in the case of surface waves. Thederived equations for U1, V 1, and ξ are expanded in powers of the wave steepness ε, and the

equations are solved to orders ε1 and ε

2 separately.

The stationary ice cover exerts a no-slip condition on the Kelvin wave motion to order

ε1. The frictional effect of the coast is neglected. So is also the frictional influence of the

lower layer. Trapped internal waves of constant frequency ω which enter our system at x=0,

will then attenuate due to the frictional effect of the ice as they propagate along the coast.

The wave velocity in the y -direction vanishes identically.

The equations for the mean fluxes are obtained by vertical integration and averaging

over the wave period 2π/ω. By calculating the nonlinear wave-forcing terms in these

equations, we find radiations stress terms for the baroclinic motion which are similar to the

radiation stress components for the barotropic case first reported by Longuet-Higgins and

Stewart (1962). Due to the spatial damping of the primary internal Kelvin wave, the mean

Lagrangian flux U1 also attenuates along the coast. In a steady state this leads to a non-zero

mean off-shore flux V 1, satisfying V 1 =0 at the coast. This result appears to be novel. The

specific form of V 1 depends on how the frictional effect on the mean flow is modeled.

The solutions from our analytical model are applied to the Van Mijen fjord in

Svalbard. Across the entrance to this fjord there is a wide island (Akseløya), preventing ice

from being transported out of the fjord by the action of the wind. Ice is thus present a large

time of the year. At the narrow and shallow entrance, barotropic tidal forcing induces

internal waves with a period of about 12.4 hours. From Berg (2004) and Widell (2006), we

infer that the fjord is much wider than the internal Rossby radius. Accordingly, internal

Kelvin waves are likely to propagate around the fjord with the coast on the right, when

looking in the wave propagation direction.

Since we have no field measurements that actually reveal the presence of internal

Kelvin waves in the Van Mijen fjord, we apply a numerical model (Gjevik 2001) to reproduce

this wave pattern numerically. The model is linear, solving for the vertically integrated fluxes

in the upper and lower layer. Boundary conditions at the open boundary are oscillations insurface and interface displacements corresponding to tidal observations in the Advent fjord



3

(Longyearbyen). Results from these model simulations demonstrate an internal wave motion

of similar character as described by our analytical analysis for a straight coast; see Fig. 1.

Fig. 1. Numerical simulation of internal Kelvin waves in the Van Mijen fjord. The figure

depicts absolute values of the wave amplitudes.

If we do assume the existence of internal Kelvin waves in the Van Mijen fjord, we

may apply the results from our second-order nonlinear analysis to this case. The calculated

along-shore and off-shore drift components within the region of trapping would then lead to

a depression of the mean interface in the interior of the basin. This would favor a quasi-

geostrophic flow in the interior which is in the opposite direction of the mean drift close to

the right-hand shore. Thus, the total wave-induced circulation under the ice in the Van Mijen

fjord appears to be anti-clockwise. In cooperation with UNIS in Svalbard field measurementsin the Van Mijen fjord are planned to test this hypothesis. In addition, a nonlinear numerical

model will be developed that can resolve the wave-induced drift in stratified fjords.

References

Berg, J., 2004: Measured and modeled tidally driven mean circulation under the ice in Van

Mijenfjorden. Master Thesis, Göteborg University.

Csanady, G. T., 1972: Response of large stratified lakes to wind. J. Phys. Oceanogr ., 2, 3-13.

Gjevik, B., 2001: LECTURES ON TIDES AT UNIS, LONGYEARBYEN, 33 pp (available on the



4

internet).

Longuet-Higgins, M.S., 1953: Mass transport in water waves. Philos. Trans. Roy. Soc.

London, A245, 535-581.

Longuet-Higgins, M.S., Stewart, R.W., 1962: Radiation stress and mass transport in gravitywaves, with application to “surf beats”. J. Fluid Mech., 13, 485-502.

Ou, H. W., and Bennett, J. R., 1979: A theory of the mean flow driven by long internal waves

in a rotating basin, with application to Lake Kinneret. J. Phys. Oceanogr ., 9, 1112-

1125.

Phillips, O.M., 1977: The Dynamics of the Upper Ocean, 2d ed. Cambridge University Press.

Stokes, G.G., 1847: On the theory of oscillatory waves. Trans. Cambridge Philos. Soc., 8, 441-

455.

Weber, J.E.H., Broström, G., Saetra, Ø., 2006: Eulerian versus Lagrangian approaches to the

wave-induced transport in the upper ocean. J. Phys. Oceanogr., 36, 2106-2117.

Widell, K., 2006: Ice-Ocean interaction and the under-ice boundary layer in an Arctic fjord ,

PhD-Thesis, University of Bergen, 65 pp.

Wunsch, C. I., 1973: On the mean drift in large lakes. Limnol. Oceanogr ., 18, 793-795.

eivind støylen and jan erik h. weber- mean mass transport induced by internal kelvin waves, with...

Documents