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 yaxis 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.
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We intend to solve the nonlinear equations of motion within an Eulerian framework.
To obtain the mean Lagrangian upper-layer volume fluxes U1 and V1 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. The
derived equations for U1, V1, and ξ are expanded in powers of the wave steepness ε, and the
equations are solved to orders ε1 and ε2 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 V1, satisfying V1 =0 at the coast. This result appears to be novel. The
specific form of V1 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 in
surface and interface displacements corresponding to tidal observations in the Advent fjord
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(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 quasigeostrophic 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 measurements
in 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
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Mijenfjorden. Master Thesis, Göteborg University.
Csanady, G. T., 1972: Response of large stratified lakes to wind. J. Phys. Oceanogr., 2, 3-13.
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