Vertical motions ABC ABC

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1 Vertical motions Determining divergence n objective method to calculate divergence from three locations with wind data has been developed by J.C. Bellamy. If wind data at multiple levels is available (e.g. from radiosondes), then we can determine the divergence at these levels. In this note we will show that you can determine vertical velocity from the divergence at multiple levels. Consider the triangle formed by the stations, B en C (Figure 1). We assume that the change of the wind between these stations is linear: we then may consider each station separately. We start with station and assume that in B and C there is no wind. We calculate the contribution to the total divergence by the wind in station. In the same way we can calculate the contributions from B and C. The total divergence then is the sum of the three partial contributions. Figure 1 Triangle of stations with wind barbs. In figure 1 ' is proportional to the wind vector station, i.e. the distance the air travels in, say, one hour. In there is no wind in B and C and the wind change between the stations is linearly then the outflow of air is represented by the area: 'C + 'B 'BC - BC The partial horizontal divergence is defined as the outflow per unit volume, hence if we consider a layer of unit thickness we have: BC BC (1) BC 1

2 Define the lines h, h ' and a as indicated in Figure 1, then we have: BC BC BC h and BC h () and for the (partial) divergence we find: h h h a h componentof thewindalongthenormal length of thenormal (3) In this expression the component of the wind along the normal is in knots and the length of the normal in nautical miles. If the position of the three stations is known we may be able to determine the length of the normal h. If also the orientation of X is known and also the wind in, then a can be calculated and hence the partial contribution of station to the divergence. In the same way the partial contributions of stations B and C can be calculated. For the total divergence we find: a b c D + B+ C + + (4) h h h B ccording to Bellamy the divergence calculated in this way applies to the centre of the triangle. ccounting for the wind direction, the component of the wind along the normal is given by: [( dd ) α] C a ff.cos 180 (5) where ff is the wind speed in m s -1, dd is the wind direction in degrees and α is the direction of the normal in degrees relative to north. We then have a in m s -1. Note that the unit of divergence D is in s -1. For the exercise we use radiosonde data from the stations Valencia (Ireland, station 03953), Long-Kesh (Northern Ireland, station 0390) and Camborne (England, station 03808). We have given (1) the wind data (in m/s) and the pre-calculated lengths (in km) and directions (from North) of the normals of the three stations, () an overview of the location of the stations and (3) a weather map of the situation we are considering. Vertical velocity ω The normal vertical velocity w (in m s -1 ) is defined in the x,y,z-system with the vertical coordinate height (in m): z w. (6) t

3 However, in the x,y,p-system with vertical coordinate pressure (in Pa) we have for the vertical velocity: ω ρgw, (7) t where the vertical velocity (ω) is expressed in the unit Pa s -1. Using hydrostatic equilibrium the relation between ω and w is easy to derive. It is almost impossible to measure the vertical velocity directly (order of magnitude is small: 10 cm/s). Hence the vertical velocity must be determined from other meteorological variables. If we know the divergence at several pressure levels, then we can calculate the vertical velocity ω. This so-called kinematic method is based on the equation of continuity in pressure coordinates: Or: u v ω (8) x y ω u v + x y D h (9) where D h is the horizontal divergence (in s -1 ). Integration between levels p 1 and p gives: p D( p1) + D( p ω( p) ω( p1) D.dpω( p1) + p p 1 ). (10) Hence if we know the divergence on two pressure levels we can calculate the vertical velocity ω on the top level if we know ω on the bottom level. Of course we start at the surface because there the vertical velocity equals zero and we use Equation (10) to calculate ω on the first level above the surface. Then we go to the next layer, and by successive integration (summation) over subsequent layers we are able to calculate the vertical velocity at each level. Order of magnitude of divergence and vertical velocity In active synoptic systems the divergence D ranges between 1x10-5 and 4x10-5 s -1. characteristic value for ω is 0-40 hpa/hour. That is approximately 1 Pa s -1 which equals w 10 cm s -1. The profile of divergence of a mature cyclone is presented in Figure. Usually we have then convergence ( negative divergence) in the lower troposphere and 3

4 divergence in the upper troposphere. Near 500 hpa we see that the divergence is almost zero: the level of non-divergence Figure Profiles of convergence and divergence. : trough in the upper troposphere, cyclone at the surface. B: developing cyclone. Divergence and vorticity The divergence is always one order of magnitude smaller than the planetary vorticity f (D 10-5 s -1 versus f 10-4 s -1 ). This is necessary because otherwise the vorticity equation is violated. This equation reads (after some scaling): 1dη η dt f 1 d( +ζ f +ζ) D. (11) dt Where η f +ζ is absolute vorticity. From this equation it appears that patterns of convergence- and divergence are strongly coupled to changes in absolute vorticity. If e.g. D < 0 (convergence), then dη/dt > 0 and we have an increase of cyclonic absolute vorticity. In reality we find that, indeed, an area with low-level convergence favours the development of a cyclone. Low pressure at the surface however, means less air in the column above it. This can be understood if we realize that there may be low-level convergence, but this may be fully compensated by upper-level divergence. In Figure 3 the relation between convergence, divergence and vertical motion is given schematically. 4

5 Figure 3. Relation between convergence, divergence and vertical velocity. 5