Environmental Engineering Reference
In-Depth Information
The governing equations, which must satisfy the above initial-boundary conditions, are
integrated with time using the simple prismatic finite elements, with the help of which basic
and trial functions are discretized. Discretization in time is produced with a semi-implicit
scheme, providing that all advective and diffusive terms in the equations for horizontal
velocity, temperature, salinity and eddy characteristics are attributed to the previous time step.
As a result, the evolutionary equations are solved at each grid node as a system of one-
dimensional (in the vertical direction) inhomogeneous ordinary differential equations. This
predetermines (to avoid numerical instability) choosing a small time step. Its value has
already been pointed out.
The model equations are integrated before the quasi-periodic state is established, which is
defined as the state when the relative changes in all components of the barotropic and
baroclinic tidal energy budgets are 5 %. If the Väisälä frequency is taken as horizontally
uniform, this condition is satisfied within 18 tidal cycles after the establishment of the quasi-
periodic regime for the M
2
barotropic tidal flow, which in turn is established from the state of
the rest within 12 tidal cycles. When the quasi-periodic state of the stratified ocean is
established, the integration is yet applied to another tidal cycle; then, it is stopped, and the
resulting solution is subjected to a harmonic analysis. In our case (the stratified ocean), the
harmonic analysis is applied only to a time series of tidal sea surface level elevations. The
time series for other dependent variables are not subjected to the harmonic analysis, so as not
to exclude any manifestations of non-linearity.
3. Model Results
In the Introduction, we have already mentioned the articles in which model results for the
surface tide in the Arctic Ocean are presented. These articles are mainly based on 2D (in a
horizontal plane) tidal models disregarding the effects of density stratification and the phase
difference between the bottom stress and the barotropic (depth-averaged) tidal velocity. From
these articles, including our one, it follows that the M
2
surface tide in the Arctic Ocean is
initiated by the Kelvin wave traveling from the North Atlantic. The interaction of this wave
with the reflected Kelvin wave, generated by partial reflection of the ingoing wave in the
Arctic Ocean, leads to the formation of the amphidrome with left (anti-clockwise) rotation of
isophases in the Central Arctic (Figures 2,3). Two more amphidromes of left rotation are
detected at the entrances to Amundsen Gulf and Davis Strait. The distributions of cotidal lines
in the straits between islands of Kanadian Arctic Archipelago differ between various
published tidal charts due to choosing different spatial resolutions and topographies. In the
adjacent seas of the Siberian continental shelf, a chain of amphidromes of left rotation is
found out. These amphidromes owe their origin to the interference of ingoing and outgoing
Poincaré waves and the predominance of the eastward wave as compared to the westward
one. The above feature conforms with the model results presented in [Androsov et al., 1998;
Kagan et al., 2008]. In this respect, our results will not change the current view of the spatial
structure of the M
2
surface tide in the Arctic Ocean.
It is interesting to compare our model results with the observational data obtained at coast
and island stations, where sea surface level elevation measurements are available (see
Figure1).
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