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Prediction and Predictability


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Figure 24. Energy spectra of the full flow (grey) and of differences between two solutions (blue) as a function of horizontal wavenumber for two-dimensional doubly periodic turbulence forced at wavenumber 8. The panel at left shows spectra for surface quasigeostrophic (sQG) turbulence, while the right panel shows barotropic turbulence. In each case, the initial differences are concentrated near $k = 20$; in the sQG case, differences grow most rapidly at the smallest scales outside the dissipation range, while for the barotropic case diferences grow at larger scales (comparable to that of the forcing for the full flow). The arguments of Lorenz (1969) suggest that the sQG flow has finite intrinsic predictability. The thin solid line segments show the theoretical power laws for the full flow of $k^{-5/3}$ and $k^{-3}$ in the sQG and barotropic cases, respectively.

The skill of today’s weather forecast is clearly limited by the accuracy of the forecast model and its initial conditions. It has long been hypothesized, however, that forecast skill in certain flows will still have an intrinsic, finite limit even given arbitrary improvements in the quality of both the forecast model and initial conditions. The validity of this hypothesis, and quantitative estimates of the intrinsic limit of predictability for different scales and phenomena, are important issues as numerical weather prediction begins to focus on quantitative precipitation forecasts (QPF) and other forecast problems with small spatial scales and short time scales.

Zhe-Min Tan (Nanjing University, China) and Fuqing Zhang (Texas A&M University), together with Chris Snyder and Richard Rotunno, have explored the limits of predictability for precipitation within the context of an idealized, moist baroclinic wave developing in a channel (Tan et al. 2004). As in previous studies by F. Zhang, C. Snyder and R. Rotunno, they find rapid growth of forecast differences in regions of (parameterized) moist convection, which then contaminate larger scales. It is clear that this growth from small to large scales places an upper bound on the skill of QPF forecasts.

As an intermediate step between studying atmospheric predictability with simple models and with complex models such as the Weather Research and Forecast Model (WRF), Rebecca Morss, C. Snyder and R. Rotunno investigated the predictability of various quasigeostrophic (QG) flows, including a three-dimensional jet in a beta-plane channel and forced two-dimensional turbulence under both barotropic and surface-QG dynamics. Using the tangent linear and full non-linear versions of the model, they examined error growth rates, the power law of the energy spectrum, and the evolution of the error spectrum during error growth. They are focusing on understanding interactions among errors in different scales and the effects on predictability, and on evaluating the long-standing hypothesis that there is a critical exponent for the energy spectrum (thought to be -3) above which flows have finite intrinsic predictability.

Error growth behavior

In the coming year, R. Morss, C. Snyder, and R. Rotunno will continue their investigation of QG predictability. They will focus on understanding the error growth behavior in the model, and on developing improved techniques for diagnosing predictability behavior that can be applied in more complex models, especially when error growth is strongly intermittent in space and time. They will then apply the results to more realistic flows to help delineate the mechanisms by which errors at the convective scale spread across the mesoscale to synoptic scales.

While the work of Zhang et al. has focused on the intrinsic limits of predictability, the practical predictability of synoptic-scale flows, given realistic observing systems, is also of interest. Analysis error statistics and their influence on forecast-error growth are crucial factors in determining practical predictability, and are poorly understood at present.

C. Snyder and Gregory Hakim (University of Washington) examine how various assumptions about analysis error statistics influence singular-vector (SV) calculations in the quasigeostrophic Eady model (Snyder and Hakim 2004). They propose an empirical covariance norm for SVs based on the assumption that potential-vorticity variance of analysis errors is small in the interior of the troposphere relative to the surface and tropopause. Their results show that a common conclusion from SV studies, namely that forecast errors arise principally from highly tilted structures in the mid- and lower troposphere, depends strongly on the assumption that the total energy norm approximates analysis-error statistics.

Next Topic: Life Cycles of Precipitating Weather Systems


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