Upper- and lower-layer separatrices in two-layer vortical pairs: (a) -plane distributed modon equilibrium achieved in a numerical simulation (see text for details); (b) typical separatrices of an -plane heton translating in the direction. Solid closed line, upper layer; dashed line, lower layer.
Notations: domains , , , and , and separatrices and . Solid closed line, upper layer; dashed line, lower layer.
Domains and , their boundaries and , and schematic of the numerical grid.
PV jump across the separatrix in the upper layer, , as a function of . -plane modons with , , and ; (a) general view with two roots 1 and 2; (b) magnified graph in the vicinity of root 1; (c) same in the vicinity of root 2.
The -plane modon given by the parameters , , , and (root 2). (a) and (b) Upper-layer contours of the comoving streamfunction and potential vorticity , respectively; (c) separatrices (solid line) and (dashed line); (d) and (e) cross sections along the axis of and , respectively; (f) vs scatter-graph. In (a) and (b), the contour interval is 10% of the maximum; dotted line is the separatrix ; shaded, regions of positive values and .
The -plane modon given by the parameters , , , and (root 1). Notations as in Fig. 5.
The supersmooth -plane modon solution given by the parameters , , , and . (a)–(f) As in Fig. 5; inset in (f), magnified view of vs scatter-graph in the vicinity of the point of contact .
The -plane modon equilibrium achieved in a numerical simulation with time integration from an unsteady initial state [Fig. 1(a)]. Notations as in Fig. 5.
Evolution of the supersmooth modon (Fig. 7) in two simulations. (a)–(c) Under perturbations caused by the numerical procedure only (including the cutting filter); (d)–(f) tilted modon. (a) and (d) in the initial states; (b) and (e) at ; (c) and (f) vs scatter graphs at .
First simulation: modon characteristics at . Notations as in Fig. 5. Changes from to are fairly small.
Second simulation: transition to a nonoverlapping state. (a) Separation between the upper- and lower-layer PV peaks vs time; (b) -component, , of the modon speed vs time.
Second simulation: modon characteristics at . Notations as in Fig. 5. The overlap of the upper and lower PV chunks is negligible.
The supersmooth -plane modon solution given by the parameters , , , and . Notations as in Fig. 7; black area in (b), zero .
Maximal difference between and , and difference between and , as functions of shift . The external parameters used are , , and .
Accuracy of the interior solution: root-mean-square (RMS) and maximal (Max) residuals. Two left columns, Eqs. (29) and (30); two right columns, Eqs. (29) and (31).
Accuracy estimates for matching of the layer streamfunctions and their derivatives at and : RMS and maximal residuals. Four left columns, conditions (33), two right columns, conditions (34).
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