Ocean Science
www.ocean-sci.net
1812-0784
1812-0792
4
4
2008
10.5194/os-4-293-2008
http://www.ocean-sci.net/4/293/2008/
http://www.ocean-sci.net/4/293/2008/os-4-293-2008.html
http://www.ocean-sci.net/4/293/2008/os-4-293-2008.pdf
293
306
2008-12-16
Retroflection from slanted coastlines-circumventing the "vorticity paradox"
V. Zharkov
D. Nof
nof@ocean.fsu.edu
Geophysical Fluid Dynamics institute, Florida State University, USA
Department of Oceanography, Florida State University, USA
The balance of long-shore momentum flux requires that the solution of
<i>zonally</i> retroflecting currents involve ring shedding on the western side. An
important aspect of the ring dynamics is the ring intensity α
(analogous to the Rossby number), which reaches its maximum value of unity
when the upstream potential vorticity (PV) is zero. Friction leads to a
slow-down and a decrease in α. The main difficulty is that the solution of
the system of equations for conservation of mass and momentum of <i>zonal</i> currents
leads to the conclusion that the ratio (Φ) of the mass flux going into
the rings and the total incoming mass flux is approximately 4α/(1+2α). This yields the "vorticity paradox" – only relatively weak
rings (α≤1/2) could satisfy the necessary condition Φ≤1.
Physically, this means, for example, that the momentum-flux of zero PV
currents upstream is so high that, no matter how many rings are produced
and, no matter what size they are, they cannot compensate for it.
<br><br>
To avoid this paradox, we develop a nonlinear analytical model of
retroflection from a slanted <i>non-zonal</i> coastline. We show that when the slant of
coastline (γ) exceeds merely 15<sup>0</sup>, Φ does not reach unity <i>regardless</i> of the
value of α. Namely, the paradox disappears even for small slants. Our slowly
varying nonlinear solution does not only let us circumvent the paradox. It
also gives a detailed description of the rings growth rate and the mass flux
going into the rings as a function of time. For example, in the case of zero
PV and zero thickness of the upper layer along the coastline, the maximal
values of Φ can be approximately expressed as, 1.012+0.32exp(−γ/3.41)−γ/225.
Interestingly, for significant slants γ≥30<sup>0</sup>), the rings reach a <i>terminal size</i> corresponding to a balance between the
β-force and both the upstream and downstream momentum fluxes. This terminal
size is unrelated to the ultimate detachment and westward drift due to
β. Our analytical solutions are in satisfactory agreement with the results of
a numerical model that we run.
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