1. Introduction

Large-scale motions in the geophysical fluids such as the atmosphere and oceans are almost horizontal and quasi-two-dimensional due to the effect of the earth's rotation and the density stratification (e.g., Cushman-Roisin,1994). The motions with a time scale longer than that of the earth's rotation are almost nondivergent and characterized by vortex (rotational) motions. Some examples are stratospheric circumpolar vortex, cut-off lows, and blocking highs in the atmosphere, and subtropical gyre, cut-off vortices from the western boundary currents, and meso-scale eddies in oceans.

Nonlinear vortex behavior such as deformation, interaction, and merging is one of the most interesting subjects in the studies on vortices in geophysical fluids (e.g., Nezlin and Snezhkin,1993). Transport and mixing processes associated with these vortex motions are also interesting. For example, these processes associated with the winter polar vortex have been investigated in detail in conection with the ozone hole problem in the atmospheric stratosphere (e.g. McIntyre,1995). The concept of eddy diffusion is not very useful in the large-scale motions because they are far from the homogeneous isotropic turbulence. Instead, new concepts such as chaotic mixing (Ottino,1989) have been introduced and the mixing process has been analyzed from kinematic point of view by using finite-time Lyapunov analysis (e.g., Pierrehumbert,1991).

In this study, nonlinear evolution of an unstable elliptical vortex patch in two-dimensional nondivergent perfect fluid is calculated by using a contour dynamics model. The results are analyzed especially from the viewpoint of mixing, and stretching of fluid element. Vortex patch has no vorticity gradient inside nor outside but has vorticity jump crossing the surrounding contour line. Contour dynamics model is a model to calculate the time evolution of vortex patches through line integrations along the contours. Our model is constracted based on Dritschel(1989)'s one, who has improved this type of model in precision and efficiency.

In generally used finite difference model and spectral models, there is a limitation in spatial resolution for phenomena such as the streamer formation accompanied with the vortex deformation, the scale of which is much smaller than vortex itself. In this point, contour dynamics model has advantages in the accurate calculation of vorticity, Jacobian matrix, streamfunction and so on at arbitrary points on the field. Furthermore, it is also the advantage that low time cost of the model facilitates various kinds of analyses.

Giving an dynamically unstable, elliptical vortex patch as an initial state, its time evolution is calculated. Deformation of the vortex and mixing of the fluid are analyzed by using streamfunction in the co-rotating frame with the vortex and finite-time Lyapunov exponents. Experiments on passive contour advections reveal the fluid motion around the vortex.

The theoretical basis of contour dynamics and analysis methed are described in Section 2.
Numerical procedure in the contour dynamics model is given in Section 3.
The result of time evolution and various analyses are shown in Section 4.
Discussions and summary in Section 5.

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