| An approach for the reconstruction of atmospheric flow is presented which uses
space- and time-dependent fields of density Ï, potential vorticity Q and potential
temperature θ [J. Phys. A, 38, 6419 (2005)]. The method is based on the fundamental
equations without approximation. The basic idea is to consider the time-dependent
continuity equation as a condition for zero divergence of momentum in four dimensions
(time and space, with unit velocity in time). This continuity equation is solved
by an ansatz for the four-dimensional momentum using three conserved stream
functions, the potential vorticity, potential temperature and a third field, denoted as
Ï-potential. In zonal flows, the Ï-potential identifies the initial longitude of particles,
whereas potential vorticity and potential temperature identify mainly meridional and
vertical positions. Since the Lagrangian tracers Q, θ, and Ï determine the Eulerian
velocity field, the reconstruction combines the Eulerian and the Lagrangian view
of hydrodynamics. In stationary flows, the Ï-potential is related to the Bernoulli
function. The approach requires that the gradients of the potential vorticity and
potential temperature do not vanish when the velocity remains finite. This behavior
indicates a possible interrelation with stability conditions. Examples with analytical
solutions are presented for a Rossby wave and zonal and rotational shear flows. |