Reduced Atmospheric Models Using Dynamically Motivated Basis Functions

Nonlinear deterministic reduced models of large-scale atmospheric dynamics are constructed. The dynamical framework is a quasigeostrophic three-level spectral model with realistic mean state and variability as well as Pacific–North America (PNA) and North Atlantic Oscillation (NAO) patterns. The study addresses the problem of finding appropriate basis functions for efficiently capturing the dynamics and a comparison between different choices of basis functions; it focuses on highly truncated models, keeping only 10–15 modes. The reduced model is obtained by a projection of the equations of motion onto a truncated basis spanned by empirically determined modes. The total energy metric is used in the projection; the nonlinear terms of the low-order model then conserve total energy. Apart from retuning the coefficient of horizontal diffusion, no empirical terms are fitted in the dynamical equations of the low-order model in order to properly preserve the physics of the system. Using the methodology of principal interaction patterns (PIPs), a basis is derived that carefully compromises minimizing tendency error with maximizing explained variance in the resolved modes. A new PIP algorithm is introduced that is more compact and robust than earlier PIP algorithms; a top-down approach is adopted, removing modes from the system one by one. The mean state and standard deviation of the streamfunction as well as transient momentum fluxes are well reproduced by a PIP model with only 10 modes. Probability density functions are accurately modeled and autocorrelation functions are captured fairly well using 15 modes. Reduced models based on PIPs are substantially superior to reduced models based on empirical orthogonal functions (EOFs). The leading PIPs have a higher projection onto the PNA and NAO teleconnection patterns than the corresponding EOFs. Both with EOFs and PIPs, the interactions between the resolved modes are predominantly linear and the improvement of PIP models on EOF models stems entirely from better modeling these linear interactions although the full nonlinear tendencies are optimized. There is considerable influence of smaller-scale modes on the large-scale modes due to nonlinear coupling that is not well captured by either EOFs or PIPs. This nonlinear backscattering possibly plays a role in generating the low-frequency variability of the model. The results call for a nonlinear and/or stochastic closure scheme in which PIPs may be suitable basis functions.