In the framework of the linear theory by using the Eady model, properties of optimal baroclinic perturbations which attain the maximum growth over a prescribed time interval (τ) are examined in connection with the skill forecast problem in numerical weather prediction (NWP). The development of the optimal perturbation depends not only on the linear stability of the basic flow, but also on the time interval τ to assess its growth.
In the unstable region, the optimal perturbation grows almost exponentially in time as the unstable normal mode. Its structure gradually coincides with the unstable normal mode as τ becomes large. The corresponding optimal initial disturbance becomes the adjoint of the unstable mode. For smaller τ (τ≤3 days), however, the phase line of the optimal perturbation has a larger inclination than the unstable normal mode, which plays a negligible role in this early development.
In the neutral region, the maximum amplitude of the optimal perturbation grows algebraically in time as Cτ²+1 (C is a constant). The optimal initial disturbance has a plane wave structure leaning against the shear. The phase line becomes more horizontal and the vertical scale decreases inversely with τ. On the other hand, the excited optimal perturbation has an almost barotropic structure, and is composed mainly of the two neutral non-singular normal modes with different phase speeds. The growth of the optimal perturbation is quantitatively well understood by the Orr mechanism for larger τ (τ≥2 days). For shorter τ, however, the interference between the two neutral non-singular normal modes has a major contribution to the development. The structure of the excited optimal perturbation is also explained by considering the projectability for each mode.
This study suggests that a high resolution model which is comparable to the operational full NWP model is necessary to predict accurately the forecast skill of the NWP model.