# JMSJ Highlights

#### Ishioka et al. (2022)

Ishioka, K., N. Yamamoto, and M. Fujita, 2022: A formulation of a three-dimensional spectral model for the primitive equations.* J. Meteor. Soc. Japan*, **100**,

https://doi.org/10.2151/jmsj.2022-022.

Early Online Release
Graphical Abstract

**Overview: **

A formulation of a three-dimensional spectral model based on the primitive equations is proposed. In this
formulation, the Legendre polynomial expansion is used for the vertical discretization. On the basis of this
formulation, a numerical model is developed and used to perform modern benchmark calculations to show
that this implementation of the primitive equations can give accurate numerical solutions with relatively
small degrees of freedom in the vertical discretization. It is also noted that semi-implicit time integration
can be efficiently done and that an alternative to the sponge layer can be devised to suppress the reflected
waves under this formulation. Hence, this formulation can be especially valuable in creating mechanistic
General Circulation Models (GCMs) for use in researches of atmospheric dynamics.

In recent years, with the improvement of computational power, nonhydrostatic atmospheric models have
become available even for the entire globe, but GCMs based on the primitive equations is still used for cal-
culations at forecast centers and for climate research. In addition to the realistic GCMs used in these fields,
mechanistic GCMs, which omit the physical processes and extract only the dynamics, are now widely used
in researches of atmospheric dynamics. The dynamical core of most GCMs has been implemented using
the spectral method with spherical harmonics expansion in the horizontal direction and the finite difference
method in the vertical direction. The use of the finite difference method in the vertical direction, however,
causes a truncation error. If a more accurate discretization is possible with the same discretization degrees
of freedom, it can lead to an improvement in computational efficiency. We propose a new formulation of
the spectral method using the Legendre polynomial expansion in the vertical direction, which avoids the
singularity at the top of the atmosphere in the expansion itself. We also describe how the semi-implicit
method can be applied under this formulation.

To check the convergence of the numerical solution with changing the vertical truncation wavenumber,
we perform benchmark numerical calculations of the growth of baroclinic disturbances. Figure 1 shows the
dependence of the l2 error (ε) of the surface pressure field on the vertical truncation wavenumber L. On days1 and 9,
the dependence of ε on L is expressed approximately as ε ∼ L^{−1}. We believe that this is caused by Lamb
wave modes directly excited by the initial disturbance. On day 11, however, in the region where L is small (L = 10
to L = 21) the L-dependence of ε is clearly higher power curve than L^{−1}(ε ∼ L^{−3}).
After day 13, when the baroclinic disturbances develop, the power of the power-law dependence becomes
higher, and on day 17, the dependence is ε ∼ L^{−6} in the range of L = 10 to L = 42. That is, the error of
the numerical solution decreases rapidly when the number of vertical degrees of freedom is increased. This
is a characteristic property of the spectral method.

One of the other advantages of using the spectral method in the vertical direction is that we can introduce an
alternative to the sponge layer by introducing the pseudo-hyper-viscosity in the wavenumber space in order
to suppress the reflected waves. Figure 2 shows the exact solutions for steady mountain wave with radiative
boundary condition and the numerical solutions computed with the pseudo-hyper-viscosity for two cases
of background wind speed U. By introducing the pseudo-hyper-viscosity, it acts like a sponge layer in the
upper atmosphere, where σ is small, and suppresses the reflected waves and a response close to the exact
solution is obtained in the lower atmosphere.