**Abstract:
**The Earth Simulator ignited high-resolution global
simulations. We
conducted 10-km simulations using AFES (AGCM for the Earth Simulator)
at T1279L96 in 2002 (Ohfuchi et al. 2004). Although we believe it was a
landmark in the history of AGCM, it seemed to me that T1279 hit the
limit of hydrostatic spectral models. In fact other groups started
developing a non-hydrostatic grid-point models such as NICAM (Tomita
and Satoh 2004) to permit explicit cumulus convection and to avoid
computationally demanding Legendre transforms. While I have enjoyed
seeing development of non-hydrostatic models, I have always felt that
spectral models can also be improved. In this seminar I will discuss my
contributions with collaborators to keep spectral models competitive in
the non-hydrostatic regime. They are 1) accurate methods to calculate
the associated Legendre functions (ALF), 2) an accurate and simple
interpolation scheme and 3) simple and stable non-hydrostatic
formulation.

ALF is the prerequisite for the accuracy of the Legendre transform thus of the dynamical core. Enomoto et al. 2008) show that ALF calculated with the common three-point recurrence fails at high degree and order and recommend the alternative four-point recurrence (Swarztrauber 1993). Although the latter enables stable calculation at high order and degree, the accuracy of each value of ALF has not been confirmed. Recently Fukushima (2011) proposed an efficient and accurate method to calculate ALF with the conventional three-point recurrence using the extended floating-point arithmetic. ALF calculated with the four-point recurrence are verified with those calculated with Fukushima’s method. ALF calculated with the four-point recurrence is accurate to about 13 digits below the floating point but are inaccurate at very small values (say <10!!"). Although these small values do not affect the accuracy of the Legendre transform, Fukusima’s method is a good replacement due to its capability to calculate wide range of values and to parallelize in the zonal wave number m.

Majority of weather forecast and some climate models uses semi-Lagrangian advection in favour of a longer time step. In addition it eliminates dispersion, which is unavoidable with Eulerian advection. The problem of semi-Lagrangian advection is dissipation inherent in interpolation. Enomoto (2008) proposes the use of the spectral derivatives in the bicubic interpolation. He conducted rotation of a Gaussian hill (Ritchie 1987) with various interpolation methods and demonstrated the advantages of the new method. Although the advective form does not formally conserve mass, dissipation is very small with this interpolation. Recently a standard suite of advection tests has been proposed (Lauritzen et al. 2012). It includes smooth and non-smooth tracers under deformational non-divergent and divergent flows. The proposed interpolation is verified with the standard tests and is found to be very competitive with other state-of-art advection schemes.

The rapid increase of computing power is making global non-hydrostatic simulations affordable. A natural approach is to extend the formulation to include the non-hydrostatic effect. The advantage of this approach is that the existing data assimilation and tools require minimal changes. ECMWF and JMA seem to pursue this approach. ECMWF has achieved TL7999 (corresponding to approximately 2.5 km) with a fast Lendre transform using the butterfly algorithm (N. Wedi, pers. comm.). Hiromasa Yoshimura (MRI/JMA) has built a non-hydrostatic version of JMA GSM using double Fourier series. Their formulations are based on Laprise 1992) that proposes the vertical co-ordinates based on hydrostatic pressure. Juang (1992, 2000) also adopts hydrostatic ℀–co-ordinates in the vertical but there are subtle differences. The latter introduces the hydrostatic temperature. In a limited-area model, such as MSM, the hydrostatic temperature may be given xternally. In a GCM, however, the hydrostatic temperature must be determined internally if is not time-independent. I investigated the two formations and found the assumption of the hydrostatic state of Laprise (1992) may be used to diagnose the hydrostatic temperature within MSM. Similarly the hydrostatic assumption of Arakawa and Konor (2009) can be used. MSM is found to run stably with any of these diagnosed hydrostatic states. The diagnosed hydrostatic temperature would enable the application of the formulation of MSM to the global domain