APRIL 2008 GERBER ET AL. 1523 Testing the Annular Mode Autocorrelation Time Scale in Simple Atmospheric General Circulation Models EDWIN P. GERBER,SERGEY VORONIN, AND LORENZO M. POLVANI Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York (Manuscript received 30 March 2007, in final form 13 August 2007) ABSTRACT A new diagnostic for measuring the ability of atmospheric models to reproduce realistic low-frequency variability is introduced in the context of Held and Suarez’s 1994 proposal for comparing the dynamics of different general circulation models. A simple procedure to compute ␶, the e-folding time scale of the annular mode autocorrelation function, is presented. This quantity concisely quantifies the strength of low-frequency variability in a model and is easy to compute in practice. The sensitivity of ␶ to model numerics is then studied for two dry primitive equation models driven with the Held–Suarez forcings: one pseudospectral and the other finite volume. For both models, ␶ is found to be unrealistically large when the horizontal resolutions are low, such as those that are often used in studies in which long integrations are needed to analyze model variability on low frequencies. More surprising is that it is found that, for the pseudospectral model, ␶ is particularly sensitive to vertical resolution, especially with a triangular truncation at wavenumber 42 (a very common resolution choice). At sufficiently high resolution, the annular mode autocorrelation time scale ␶ in both models appears to converge around values of 20–25 days, suggesting the existence of an intrinsic time scale at which the extratropical jet vacillates in the Held and Suarez system. The importance of ␶ for computing the correct response of a model to climate change is explicitly demon- strated by perturbing the pseudospectral model with simple torques. The amplitude of the model’s response to external forcing increases as ␶ increases, as suggested by the fluctuation–dissipation theorem. 1. Introduction HS94 benchmark, one that characterizes the variability of a model on lower frequencies. This new diagnostic Held and Suarez (1994, hereinafter HS94), estab- establishes the fitness of a GCM for the very long in- lished a benchmark for comparing the numerical tegrations that are typically needed for studies of low- schemes of different dynamical cores, general circula- frequency variability. tion models (GCMs) that integrate the primitive equa- The computational efficiency of GCMs with the tions with idealized physics. They proposed a simple set HS94 forcing, or other similar idealized forcings, has of forcings that produce a realistic climate without com- made them an attractive tool for dynamical studies of plex parameterizations, allowing a comparison of the atmospheric variability on intraseasonal and interan- dynamical fidelity of GCMs independently of differ- nual time scales. They have been used to explore strato- ences in their radiation, convection, and boundary layer spheric–tropospheric interactions (Taguchi et al. 2001; schemes. HS94 suggested that two key diagnostics be Polvani and Kushner 2002; Taguchi and Yoden computed: the time and zonal mean zonal winds, and 2002a,b; Taguchi 2003a,b; Song and Robinson 2004; the time and zonal mean temperature variance. To- Kushner and Polvani 2004, 2005, 2006; Eichelberger gether, these diagnostics ensure that a core’s dynamics and Hartmann 2005), and to understand the behavior produce a realistic 1) mean climate and 2) synoptic ed- of the North Atlantic Oscillation, annular modes, and dies. In this paper, we propose a new diagnostic for the zonal index (e.g., Robinson 1996; Franzke et al. 2004; Son et al. 2008). In addition, a well-defined forcing with less reliance on subgrid-scale parameterizations enables Corresponding author address: Edwin P. Gerber, Dept. of Ap- plied Physics and Applied Math, Columbia University, 200 S. W. greater control over the climatology. Theories can be Mudd Building, MC 4701, 500 W. 120th St., New York, NY 10027. tested over a wide range of parameter settings (e.g., E-mail: [email protected] Schneider 2004; Walker and Schneider 2006), a strategy DOI: 10.1175/2007MWR2211.1 © 2008 American Meteorological Society MWR2211 1524 MONTHLY WEATHER REVIEW VOLUME 136 that cannot be implemented with more comprehensive cycle and ocean processes, which are often not repre- GCMs without retuning parameterizations. sented in simple GCMs. On these time scales, the lead- In a recent study with a model nearly identical to the ing patterns of variability in the extratropics are the pseudospectral dynamical core used by HS94, Gerber well-known annular modes (Thompson and Wallace and Vallis (2007) noted a sensitivity of the low- 2000). The northern and southern annular modes are frequency variability to the model’s resolution. The defined as the first empirical orthogonal functions model, however, consistently produced a realistic cli- (EOFs) of sea level pressure or geopotential height in mate by the standards of the HS94 diagnostic. This sen- the Northern and Southern Hemispheres, respectively. sitivity poses a problem for the interpretation of studies The observed annular mode patterns reveal a robust with idealized GCMs, particularly if the processes in hemispheric-scale dipole in pressure, or mass, between question depend on the nature of the low-frequency the high and midlatitudes that, by geostrophy, charac- variability. Paradoxically, Gerber and Vallis (2007) ob- terize vacillations in the position of the extratropical served a worsening of their model’s ability to capture jets. In this respect the annular modes are related to the realistic low-frequency variability with an increase in zonal index, a measure of the zonal circulation first the vertical resolution, also raising questions concern- investigated by Rossby (1939), and refined by Namias ing numerical convergence. Boer and Denis (1997) pro- (1950). Given near-geostrophic balance on large spatial posed a test for the numerical convergence of a dynami- scales, there is a very high degree of correlation be- cal core in terms of the mean flow and synoptic vari- tween the time series associated with the annular mode ability, but the issue of convergence has not been patterns and the zonal index, especially in idealized investigated in the context of low-frequency variability. models. While other, more regional patterns of variabil- To address these concerns, in this paper we first pro- ity, such as the North Atlantic Oscillation, are also pose and then apply a simple test designed to assess a prominent in the atmosphere, Feldstein (2000b) shows model’s ability to produce realistic low-frequency vari- that the time scale tends to increase with the zonal scale ability. As shown in section 2, the test involves compu- of the pattern: the broader the pattern in longitude, the tation of a single number ␶, the autocorrelation time slower the time scale. Hence a characterization of the scale of the annular mode. The procedure for comput- annular mode time scale establishes an upper bound on ing ␶ is described in section 2a. In section 3, we inves- the low-frequency variability in a model. tigate the dependence of ␶ on model numerics using For our test, we define the annular mode to be the 1 two dynamical cores, one pseudospectral and the other first EOF of the zonally averaged surface pressure ps , finite volume, forced with the HS94 scheme. From a corrected for latitude bias as in North et al. (1982). The theoretical perspective, we ask the following question: advantage of this variable is that, in the absence of What is the actual autocorrelation time scale for the topography, it requires no vertical interpolation from physical system described by the dry primitive equa- the actual model levels, as would be the case if, for tions driven by the HS94 forcing, and how does this instance, the 250-hPa zonal winds were used. Thomp- time scale compare to the one observed in the atmo- son and Wallace (2000) show that patterns identified sphere? From a practical standpoint, we wish to deter- from zonally averaged variables are equivalent to those mine what resolution is sufficient for these models to be identified from the two-dimensional latitude–longitude used to study low-frequency variability, and to provide fields, so that little information with respect to the an- a benchmark for comparison with other dynamical nular mode is lost by first averaging ps around the lati- cores. In section 4, we illustrate a potential conse- tude circle. The principal component time series asso- quence of poorly simulated low-frequency variability: ciated with the annular mode EOF pattern is denoted erroneous response to climate change. Our conclusions the “annular mode index.” We quantify the behavior of are found in section 5. the annular mode by computing ␶, the e-folding time scale of the annular mode index’s autocorrelation func- 2. The autocorrelation time scale of the annular tion. As seen in Fig. 1, such autocorrelation functions e mode are not perfectly exponential, but the -folding time scale provides a fair estimate for comparison. We seek a simple diagnostic to determine whether a It is worth emphasizing that ␶ is not an artificial sta- model exhibits realistic low-frequency variability. By tistical construction: its value has physical significance low-frequency, we mean intraseasonal, that is, time scales greater than 10 days, and so longer than the pe- riods associated with baroclinic instability, but gener- 1 If the model includes topography, zonally averaged sea level ally shorter than those associated with the seasonal pressure should be used instead of ps . APRIL 2008 GERBER ET AL. 1525 tem’s internal dynamics, the other being the eddy turn- over time scale, which describes the synoptic variability associated with baroclinic eddies. Therefore, as part of model validation, it is important to determine whether the HS94 system of equations, solved with a specific numerical scheme running at a particular resolution, generates a value of ␶ is comparable to the one ob- tained from observations.