from the closed part of the trajectory and is carried away by a stream along one of
the separatrix “whiskers”.
A small displacement of the initial position of the lens center inside the separatrix loop (Fig. 6b, where the initial coordinates of the center are indicated in the
caption) leads to the fact that now the lens makes three revolutions inside the loop
along an insignificantly untwisting spiral, and then, like the previous case, is carried
away by flow beyond the depression. Finally, if the displacement of the initial
position of the lens center toward the center of the topographic vortex becomes
larger (Fig. 6c), the lens is completely captured by the topography. At the end of the
calculation time interval, the lens performed 10 complete cycles. This series of
calculations shows that the region of the lens capture by the depression occupies
almost the entire separatrix loop of the current function except for a small neighborhood near the hyperbolic singular point. It is important to note that the capture
can take place even in the case when the lens is originally located outside the
depression (the figure in panel “c” shows this clearly). Thus, the trajectory of the
lens center in panel “c” can be approximately taken as the outer contour of the cross
section (in the middle layer) of the Taylor column [7, 9, 25], which limits the region
of the initial positions of the lens centers that do not leave the vicinity of the
depression.
The unperturbed separatrix of the phase portrait of the middle layer is shown in
all three panels of Fig. 6. However, it is obvious that the presence of a lens (if it is
not initially placed at a stationary elliptic point, as in Fig. 5) substantially alters the
entire structure of the flow and the configuration of the separatrices in particular.
Figure 7 shows how a lens affected by an “external field” changes the field of the
current function over the depression and at its vicinity. Here, we show a sequence
of instantaneous lens configurations in the case presented in Fig. 6a, when, at the
initial moment, the lens center was located at the hyperbolic point of the unperturbed separatrix. In all panels, the stationary unperturbed separatrix is depicted by
a grey solid line. It is not related to the real non-stationary velocity field and is only
a marker curve that tracks the motion of the lens center. The first panel shows that
already at t = 0, two separatrices begin their formation (red and black solid lines) in
the current function field. Note that, theoretically, at this stationary point they
should merge, but as mentioned above, the coordinates of the hyperbolic point are
calculated with an error, and therefore two separatrices have a small relative displacement (the black line has a self-intersection point located to the southwest of the
lens center, and red one to the northeast). This experiment also gives an opportunity
to demonstrate the uneven motion of the lens along its trajectory. Indeed, since the
starting position is connected with a stationary point of the phase portrait, at the
initial stage of the movement, the lens moves very slowly (the second panel corresponds to time t = 8). Then the motion accelerates, and the maximum velocity of
the lens along the trajectory occurs at the maximum distance from the initial
position. In the course of the lens re-approach to the hyperbolic point of the
unperturbed separatrix, its motion slows down again (see indicated times at the
panels). Note that a similar effect appeared in the calculations by Köhl et al. [8]
(see Fig. 11).
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B. N. Filyushkin et al.
the separatrix “whiskers”.
A small displacement of the initial position of the lens center inside the separatrix loop (Fig. 6b, where the initial coordinates of the center are indicated in the
caption) leads to the fact that now the lens makes three revolutions inside the loop
along an insignificantly untwisting spiral, and then, like the previous case, is carried
away by flow beyond the depression. Finally, if the displacement of the initial
position of the lens center toward the center of the topographic vortex becomes
larger (Fig. 6c), the lens is completely captured by the topography. At the end of the
calculation time interval, the lens performed 10 complete cycles. This series of
calculations shows that the region of the lens capture by the depression occupies
almost the entire separatrix loop of the current function except for a small neighborhood near the hyperbolic singular point. It is important to note that the capture
can take place even in the case when the lens is originally located outside the
depression (the figure in panel “c” shows this clearly). Thus, the trajectory of the
lens center in panel “c” can be approximately taken as the outer contour of the cross
section (in the middle layer) of the Taylor column [7, 9, 25], which limits the region
of the initial positions of the lens centers that do not leave the vicinity of the
depression.
The unperturbed separatrix of the phase portrait of the middle layer is shown in
all three panels of Fig. 6. However, it is obvious that the presence of a lens (if it is
not initially placed at a stationary elliptic point, as in Fig. 5) substantially alters the
entire structure of the flow and the configuration of the separatrices in particular.
Figure 7 shows how a lens affected by an “external field” changes the field of the
current function over the depression and at its vicinity. Here, we show a sequence
of instantaneous lens configurations in the case presented in Fig. 6a, when, at the
initial moment, the lens center was located at the hyperbolic point of the unperturbed separatrix. In all panels, the stationary unperturbed separatrix is depicted by
a grey solid line. It is not related to the real non-stationary velocity field and is only
a marker curve that tracks the motion of the lens center. The first panel shows that
already at t = 0, two separatrices begin their formation (red and black solid lines) in
the current function field. Note that, theoretically, at this stationary point they
should merge, but as mentioned above, the coordinates of the hyperbolic point are
calculated with an error, and therefore two separatrices have a small relative displacement (the black line has a self-intersection point located to the southwest of the
lens center, and red one to the northeast). This experiment also gives an opportunity
to demonstrate the uneven motion of the lens along its trajectory. Indeed, since the
starting position is connected with a stationary point of the phase portrait, at the
initial stage of the movement, the lens moves very slowly (the second panel corresponds to time t = 8). Then the motion accelerates, and the maximum velocity of
the lens along the trajectory occurs at the maximum distance from the initial
position. In the course of the lens re-approach to the hyperbolic point of the
unperturbed separatrix, its motion slows down again (see indicated times at the
panels). Note that a similar effect appeared in the calculations by Köhl et al. [8]
(see Fig. 11).
342
B. N. Filyushkin et al.