Fig.1

Schematic illustration of the relation between contour length of a tracer and mixing efficiency. (a) Minimum possible contour length on a sphere (latitude circles) which is at the lowest mixing efficiency; (b) smooth climatological contour through a Eulerian time mean; (c) instantaneous wavy contour distribution in which mixing efficiency is greatly enhanced. Note that the areas between any two contours are the same for each panel"

Fig. 2

Evolution of a passive tracer driven by surface geostrophic flows. The tracer is initially aligned with latitudes and normalized between 1 and 2"

Fig. 3

(a) Minimum possible length ${{L}_{\text{min}}}$ (km, blue) and total length of latitudinal circle (km, orange), and (b) the ratio of ${{L}_{\text{min}}}$ to the total length of latitudinal circle"

Fig. 4

Tracer horizontal distribution and its contour length L and equivalent length ${{L}_{\text{eq}}}$. Left columns are tracer distributions with several contours highlighted. Right columns are contour lengths normalized by ${{L}_{\text{min}}}$(b, d, f, h). The first row is the results at first day (a) and the last three rows are results after 1-yr integration with small-scale diffusivity ${{\kappa }_{\text{m}}}=$ 5(c), 20(e), 50 m2·s-1(g)"

Fig. 5

Temporal evolution of (a) equivalent length ${{L}_{\text{eq}}}$ normalized by ${{L}_{\text{min}}}$, (b) perimeter length L normalized by ${{L}_{min}}$, and (c) ratio of ${{L}_{\text{eq}}}$ to L"

Fig. 6

Temporal evolutions of (a) mixing efficiency ${{M}_{\text{E}}}$ (b) perimeter length factor ${{L}^{2}}/L_{\text{min}}^{2}$, and (c) gradient factor $\overline{\left| \nabla q \right|}\overline{{{\left| \nabla q \right|}^{-1}}}$"

Fig. 7

Comparison of Eulerian zonal mean eddy flux and along-contour mean diffusive flux. (a) Eddy tracer flux ${{\overline{v'q'}}^{X}}$ (10-5 color shadings) and ${{\bar{q}}^{X}}$ (contours) based on Eulerian zonal mean ${{\overline{\left( \text{ }\!\!\cdot\!\!\text{ } \right)}}^{X}}$; (b) diffusive flux ${{K}_{\text{eff}}}\partial q/\partial {{\varphi }_{\text{eq}}}/a$ (10-5) and $q\left( t,{{\varphi }_{\text{eq}}} \right)$ based on the along-contour mean. Red lines indicate zero-flux contour"

Fig. 8

Tracer distributions at different timesteps (a, c, e, g) and contour lengths normalized by ${{L}_{\text{min}}}$(b, d, f, h). Row one to row four show the results of integration after 1, 20, 100, and 365 days. Contour lengths are calculated using ‘box-counting’ method with 8 grid boxes"

Fig. 9

Contour length (m) against grid resolution (m) on a log-log plot for the equivalent latitude of 53.3°S. Panels (a) to (i) are results at 0, 10, 50, 100, 150, 200, 250, 300, and 350 days, respectively. Linear-fit slopes are indicated in each panel. Purple shadings indicate 95% confidence intervals for the linear fit. Two dash lines are the fitted lines with maximum and minimum slopes during the time range of interest"

Fig. 10

Spatio-temporal variations of the “box-counting” fractal dimension"

Fig. 11

Scatter plot of fractal dimension ${{D}_{\text{min}}}$ and mixing efficiency ${{M}_{\text{E}}}$. Panels (a) to (i) are results over different latitude bands. Colors indicates the time (day) relative to the first day. The black lines indicate a least-square linear fit, with residual R and the fitted equation labeled in each panel"

Tab. 1

Linear fitted coefficients and their standard errors in figure 11 for different latitude bands"