Loss of Balance


Loss of balance involves the transfer of energy from the geostrophic flow to ageostrophic modes such as inertia gravity waves that can more readily transfer energy forward in wavenumber space towards dissipation scales.  A simpler analog is to consider energy exchanges between two-dimensional and three-dimensional modes of nearly 2-d turbulence.  

Straub, D. N. 2003, Instability of 2D flows to hydrostatic 3D perturbations, J. of the Atmos. Sci., 60, pp. 79-102.

This paper points out that random strain arguments typically evoked to show that material line elements grow can also be used to argue that thin-aspect ratio 3d perturbations to 2d turbulence also grow.  This growth, however, is much reduced at small Rossby number.  The paper also generalizes some classic instability conditions.  For example, conditions for inertial instability of parallel plane flow and centrifugal instability of a circular vortex are shown to be limits of a single criterion involving the strain, vorticity, Coriolis parameter, and rotation rate of the strain axis.

With K Ngan and P Bartello, we considered 3-dimensionalization of decaying 2d turbulence in a triply periodic Boussinesq setting.  Although the instabilities first appear visually as small ribbed structures, the transfer was found to be primarily into large horizontal and small vertical scales.  This energy transfer continued long after the 3d energy reached saturation levels, and provided a weak sink of 2d energy:  the return to isotropy was slow.  We also found the saturation level of 3d energy to be dependent on the domain vertical-to-horizontal aspect ratio, with thin domains implying low 3d saturation levels.

Ngan, K., D. N. Straub and P. Bartello, 2005, Aspect ratio effects in quasi-2D tur- bulence. Phys. of Fluids. Vol 17, #1, pp. 1-17.

Ngan, K.* D. N. Straub and P. Bartello, 2004, Three-dimensionalization of freely- decaying two-dimensional turbulence, Physics of Fluids Vol 16, #8, pp. 2918-2932. * Research Associate under my co-supervision.

We did a similar study examining the instability of geostrophic turbulence to ageostrophic perturbations.  A Boussinesq code was initialized with homogeneous mature quasigeostrophic turbulence, which then evolved according to the Boussinesq equations.  A very quick growth of wave modes (consistent with an adjustment to higher order balance) occurred and was followed by a slower by more substantial growth of wave (non-quasigeostrophic) energy.  As above, this geostrophic-to-wave energy transfer persisted long after the wave energy saturated -- thus providing a weak sink of energy for the geostrophic flow.  (An appendix details the equivalence between quasigeostrophy and the absence of wave modes in the wave-vortex decomposition of the Boussinesq equations; a similar proof for the shallow water equations is found in Salmon's book on GFD.)

Ngan, K., P. Bartello and D. N. Straub, 2008, Dissipation of synoptic scale flow by small scale turbulence, J. Atmos. Sci., Vol 65, #3, 766-791.

We also looked at implications of these ideas for predictability; revisiting the classic work of Leith and Kraichnan in the context of rotating-stratified flows.

Ngan, K., P. Bartello and D. N. Straub, 2009, Predictability of rotating stratified turbulence, J. Atmos. Sciences, Vol. 66, pp. 1384-1400.