Nilawar, R.S.; (2012) Effects of finite Rossby radius on vortex-boundary interactions. Doctoral thesis , UCL (University College London).

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Abstract

The effect of the finite Rossby radius on vortex motion is examined in a two-dimensional inviscid incompressible fluid, assuming quasigeostrophic dynamics in a single layer of fluid with reduced gravity for two geophysically significant problems: a vortex near a gap in a wall and a pair of steady translating vortices. For the motion of a point vortex near a gap in an infinite barrier, a key parameter determining the behaviour of the vortex is a, the ratio of the Rossby radius of deformation and the half-width of the gap. For large a, depending on the location of the vortex, a vortex sheet is placed either over the gap (gap method) or over the two semi-infinite barriers (barrier method). When the vortex sheet is over the gap, numerical inaccuracies are encountered when the vortex is close to the gap, therefore the conjugate (barrier) method is used. Both integral equations contain singularities which can be de-singularised and solved iteratively using the known exact solution in rigid-lid limit, i.e. a → ∞. For large a, there is only slight deviation from the analytical (a → ∞) trajectories. For smaller a, the integral equation from the conjugate method is solved by numerically approximating the integral equation into a system of linear equations and solving using matrix inversion. The integral equation is further simplified by splitting into even and odd parts, thus reducing the problem to the half plane. It is also found that decreasing a, increases the tendency for vortices to pass through the gap. Background flows influence vortex trajectories and are incorporated by modifying the conjugate method integral equation. These equations are solved using the matrix method. Streamlines for uniform symmetric and anti-symmetric (which has no analogy in the rigid-lid limit) flow through the gap are computed and their effect on the vortex trajectories are found. The motion of finite area patches of constant vorticity near a gap in a wall is computed using the matrix method in conjunction with contour dynamics. For fixed a, vortex patches are normalised to travel at the same speed as a point vortex. The normalisation is non-trivial and depends nonlinearly on the patch area and a. In the rigid-lid limit, it reduces to the ‘usual’ normalisation based on the patch circulation. For near circular patches, the trajectory of the centroid of the patches also follows the trajectory of the point vortices. When the patch becomes distorted the agreement is not so close. The splitting and joining of contours is also computed using contour surgery and some examples showing this sudden change of behaviour is presented. The next problem determines the effect of the Rossby radius of deformation, on steady translating vortex pairs or, equivalently, a patch in steady translation near a wall. The velocities for the normalised vortex patch are compared to the velocity of a point vortex located at the centroid of the patch. It is found there is good agreement for a range of patch sizes. When the patches are sufficiently far from the wall, decreasing the Rossby radius makes the steadily translating shapes more circular. However, when close to the wall, the effect of the Rossby radii results in patches deforming greatly, forming long slug-like shapes. These are shown to be stable using a time dependent contour dynamics code. Background flows are also incorporated and give different vortical shapes for finite Rossby radii flows, ranging from slug-like to tear-drop in shape.

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