Figure 1.
Schematic illustration of the interaction of three point vortices arranged according to the atmospheric Omega pattern, where the circles indicate the direction and relative strength of rotation. The dotted arrows represent the influence of the other two vortices on the velocity of the corresponding point vortex. Their vector addition given by the solid lines represents the resulting velocity vector for the corresponding vortex. The anti-cyclonic vortex (red) is assumed to be twice as strong as the cyclonic vortices (blue); therefore the induced velocity field is stronger. This interaction can also be derived from Equations 2.
Figure 2.
(Left) Two exemplary blocking events, one resembling an Omega (top) and the other a High-over-low (bottom). Shown are the vorticity (coloured) and the geopotential height isolines (grey isolines in 8 dm intervals, bold line represents the 552 dm line) at 500 hPa. (Right) Illustration how the corresponding blocking can be realized in the point vortex model. Upper right figure by courtesy of Müller et al. (2015).
Figure 3.
Schematic diagram and comparison of the set-ups of the two different approaches of the contour method and the trapezoid method to determine the vortex properties. Grey-shaded boxes indicate the temporally averaged fields over one blocking period.
Figure 4.
Composites of (a, b) High-over-low blockings and (c, d) all Omega blockings that were identified by the (a, c) contour and (b, d) trapezoid method from 347 blockings during 1990–2012. The mean positions and circulations (in ) of the identified blocking vortices are marked (circles, rectangles and triangle). The field is shown for the contour method and the field for the trapezoid method. Also, the geopotential height field is shown as grey isolines in 8 dm intervals at 500 hPa, where the bold line represents the 552 dm line.
Figure 5.
Histogram of the distances l between the vortices (a-d) and the circulations (e-h) of the single-time steps for High-over-low and Omega blocking as analysed with the (a,c,e,g) contour method and (b,d,f,h) trapezoid method. Due to overlapping distributions, the colours accordingly appear darker.
Figure 6.
Scatter plot of the circulations (averaged for each blocking period) for comparing the two methods. Only situations have been considered, where both methods classify the period either as High-over-low or as Omega blocking. The dashed line shows the ideal case, the bisecting line. The correlation coefficients are displayed in corresponding colours.
Figure 7.
Scatter plot of the velocities and with the zonal mean zonal velocity averaged over . The grey-dashed line indicates the bisecting line, the blue line shows the linear regression.
Figure 8.
Phase space of relative distances . The fixed point is marked as red cross and the three eigenvectors are displayed as green (stable), blue (unstable) and grey (neutral) lines. Three exemplary trajectories are displayed as points. The elapsed time between two consecutive points corresponds to 8 h. The initial condition is marked as star in corresponding colour. Note that the grey trajectory lies on the neutral eigenvector at the initial position and is therefore stationary.
Figure 9.
Simulations of two point vortex systems applying realistic atmospheric conditions. The initial triangles (1) are disturbed from the relative equilibrium of the equilateral triangle of side length km. The distance between the two lows is (a) decreased with km, (b) increased with km. The coloured lines mark the trajectories of the corresponding point vortices. Some exemplary triangle constellations 1–6 as realized in the simulations are added for the following times: (a) days; (b) for days. When they appear after the equilateral triangle constellation (constellation 1 and 4) and before the trilinear constellation (constellations 3 and 5), the triangles are changed according to the unstable direction, as e.g. constellation 2. Triangles, changed in the stable direction exist after the trilinear constellation and before the equilateral triangle constellation, as e.g. constellation 6.
Figure 10.
Intervortical distances of the point vortex system of an exemplary simulation with friction as in Zhu and Cheng (2010). Initial set-up of the distances was km. Random numbers were drawn from a normal Gaussian distribution of zero mean and standard deviation km using R function set.seed(12345) in order to estimate the Brownian motion. The other initial conditions are described in the text.
Table 1.
Overview of the number of blocking periods as classified by the two methods. The last column gives the number of blocking periods, that were identified as either High-over-low or Omega by both methods.
Table 2.
Results of the multiple linear regression for Omega blocking. The values show the coefficients of the linearized point vortex equations and the values denote the estimates from the linear regression. Small p-values indicate more significant regression estimates. For the contour/trapezoid method, the regression models yield an adjusted of 0.28/0.50.
Table 3.
As Table 2, but for High-over-low situations. The adjusted of the regression model are 0.39 and 0.48 for the contour and trapezoid method respectively.