# Idealized Global Spectral Atmospheric Models

A global spectral atmospheric model decomposes the flow into spherical harmonic components. It provides an elegant algorithm for atmospheric modeling on global scales. It is not the algorithm currently favored for comprehensive climate modeling at GFDL, due to the difficulty of exactly conserving total mass of tracers and of dry air, and due to problems associated with Gibbs’ ripples created by trying to represent the Earth’s topography with a finite set of spherical harmonics. However, the spectral model continues to be useful for research with idealized models and for education.

### Barotropic model

**Basic barotropic model**The barotropic model solves the vorticity equation for the evolution of a two-dimensional non-divergent flow on the surface of a sphere. Default setting generates the free evolution of an eddy of a given zonal wavenumber on a stable mid-latitude zonal jet, as in Held and Phillips. Optionally, two passive tracers may be included, one advected with the spectral algorithm and another advected with a piecewise linear finite-volume scheme.

**Barotropic model with random stirring**An alternative setting is available that illustrates how random stirring can create zonal jets., following Vallis,G.K., E.P.Gerber, P.J.Kushner, and B.A.Cash, 2004:A Mechanism and Simple Dynamical Model of the North Atlantic Oscillation and Annular Modes Journal of the Atmospheric Sciences, 61(3), 264-280.

For a full description of the model and algorithms used, see The barotropic vorticity equation

### Shallow water model

**Basic shallow water model**The shallow water model solves for the evolution of a uniform density, incompressible flow on a sphere in the hydrostatic approximation (valid when the horizontal scale of the motion is large compared to the depth of the fluid). The response to heating/cooling in the atmosphere in such a model is mimicked by specifying mass sources/sinks. Default mass sources/sinks provide relaxation of height to a profile which has a ridge at the equator and an isolated hump in mid-latitudes, representing heating in the intertropical Convergence Zone and monsoonal heating, respectively, with background radiatie cooling.

For a full description of the model and algorithms used, see The shallow water equations

### Dry spectral dynamical core for hydrostatic flow of an ideal gas

**HSt42**The model setup follows the standard described in Held,I.M., and M.J.Suarez, 1994: A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models Bulletin of the American Meteorological Society, 75(10), 1825-1830. Several default settings are provided. for running in climate mode (force, dissipative flow in which one is interesting in the long term behavior, independent of initial conditions) and in initial value model (idealized initial conditions, illustrating the development of midlaittude cyclones)

**Polvani_2004**sets up the initial value problem describe in Polvani, L. M., R. K. Scott, and S. J. Thomas, 2004: Numerically Converged Solutions of the Global Primitive Equations for Testing the Dynamical Core of Atmospheric GCMs Mon. Weather Rev., 132, 2539-2552.**Jablonoski_2006**is an alternative initial value problem of the same style: Jablonowski, C. and D. L. Williamson, 2006: A baroclinic instability test case for atmospheric model dynamical cores Q.J.R. Meteorol. Soc., 2006, 132, 2943-297**Polvani_2007_LC1**adds a detailed specification of idealized tracers in the environment of a developing baroclinic wave: Polvani, L. M. and J. G. Esler, 2007: Transport and mixing of chemical air masses in idealized baroclinic life cycles J. Geophys. Res., 112, D23102, doi:10.1029/2007JD008555.

For a full description of the model and algorithms used, see Equations and numerics of the spectral dynamics code