A New Theory for the Atmospheric Energy Spectrum

The horizontal spectra of atmospheric wind and temperature at the tropopause have a steep -3 slope at synoptic scales, but transition to -5/3 at wavelengths of the order of 500-1000 km (Nastrom and Gage, 1985). Moreover, more recent analyses indicates a forward cascade of energy on scales of the order 100km (Cho and Lindborg, 2001). We demonstrate that such behavior is attainable in a quasigeostrophic model when surface temperature dynamics are included.

In the simplest example, a fluid possessing constant potential vorticity and stratification over a finite depth, with temperature perturbations generated by large-scale forcing at the upper surface and isothermal conditions at the lower surface, gives rise to a forward cascade of energy at the upper surface. The transition from a large scale -3 slope to a small scale -5/3 slope occurs at horizontal wavenumber k_t = f/NH, where f is the Coriolis parameter, N is the stratification frequency, and H is the depth of the fluid. [The figure below shows spectra generated as resolution is varied for fixed k_t, and as k_t is varied for fixed resolution, using this model.] Using atmospheric values gives a scale close to but somewhat larger than the observed transition scale. Moreover, in this example flow the forcing is artificial.

A more realistic representation of large-scale atmospheric turbulence involves baroclinic forcing by an unstable mean shear. We show that by including surface temperature advection dynamics in a baroclinically-forced stratified quasigeostrophic flow with sufficient vertical resolution to represent the temperature signals at the submesoscale, one can again obtain a forward energy cascade at surfaces or levels of sharp jumps in the stratification. Sufficient vertical resolution is the key: since temperature is proportional to the vertical derivative of the streamfunction, only those temperature-dominated motions at a wavenumbers k < f/N\Delta z are properly represented in a numerical model.

Finally, we show that sufficiently strong forcing by the interior dynamics can push the transition scale (from spectral slope -3 to -5/3) to higher wavenumbers, resulting in spectra that closely resemble the observed spectra of atmopsheric turbulence.

References:

1. Nastrom, G. D. & K. S. Gage: 1985. J. Atmos. Sci., v.42, 950-960

2. Cho, J.Y.N. & E. Lindborg: 2001. J. Geophys. Res., v.106, 10223-10232