Connections:

BjerkenesÕ Circulation Theorem is

(2a,b)

(If absolute circulation is conserved, then (2a,b) can be manipulated to obtain the simplified vorticity equation).

(2c)

The synoptically-scaled equation of continuity gives the SURFACE PRESSURE TENDENCY EQUATION:

(2d)

where is the NET HORIZONTAL DIVERGENCE in the air column above a given level. where the linkup between layer weighted horizontal divergence and vertical motion comes from the the continuity equation

(2e)

The Sutcliffe-Petterssen Development Equation basically gives the linkup between divergence patterns in the upper troposphere related to vorticity advection (the divergence due to the shape of the pattern at any given time) and temperature changes (the divergence due to isallobaric changes related ot thickness rises or falls in the upper troposphere) and is

(2f,.g)

where C is the motion of the system. Thus the effect of the synoptically-scaled divergence term in equations (2a,b,c,d and e) on the development of surface systems can be approximated by (2f, g). Remember that this is synoptically scaled.

If one assumes that motion of the system is small compared to the 500 mb geostrophic wind, then equation (b) says that the 1000 mb geostrophic relative vorticity tendency can be computed from the 500 mb vorticity advection (assuming that the 500 mb wind and vorticity fields are geostrophic) and a term that is proportional to the 1000-500 mb thickness tendency.

Thus the vorticity tendency (a prognostic quantity) could be estimate from charts at ONE synoptic time if one could change the thickness tendency to a diagnostic term. In fact, substitution of the hypsometric relation into the thickness tendency term transforms that term (with additional synoptic-scaling) proportional to the negative of the Laplacian of the temperature advection.

The equation states that 1000 mb relative vorticity will increase when there is positive vorticity advection at 500 mb and/or if thicknesses increase above the surface location in question (can be assessed from the temperature advection at a given time -- warm advection produces thickness rises).