Empirical spectral relationship of potential temperature and streamfunction on the tropopause

Surface quasi-geostrophic (SQG) dynamics on the tropopause comprise a useful description of upper level disturbances. The resemblance of the SQG dynamics with the dynamics of the two-dimensional flows renders them familiar and tractable. The pivotal relationship of SQG is the one between the potential temperature (&theta) of the tropopause and streamfunction (&psi).

In SQG, the amplitudes of spectral components of &theta are proportional to those of &psi, where the coefficient of proportionality is a function of the two-dimensional wavenumber (in 2D flows the proportionality constant is the square of the wavenumber.)

Here we use ERA40 reanalysis data on the dynamical tropopause. We spectrally analyze &theta and &psi on the surface of 2 units of potential vorticity, commonly taken as the tropopause. A regression in time for each of the spherical harmonic components of the two fields is then calculated. These linear relations are compared with the SQG relation to identify any departures in the data from the assumptions of quasi-geostrophic dynamics.