ishioka(at)gfd-dennou.org

Computational Geophysics, Computational Geophysics–Excersize, Meteorology I, Meteorology II, Laboratory Work in Earth & Planetary Sciences DD(shared)

I have been conducting theoretical and numerical studies mainly based on our interest in geophysical fluid dynamics in order to deepen our understanding of atmospheric motions on the earth and planets. I have also been conducting research related to the development of numerical methods necessary for these studies.

The following is a summary of my main research topics.

Research on pattern formation from turbulence and convection

I have been studying the pattern formation from turbulence in rotating systems with the motivation of elucidating the origin of the striped structure of zonal flows such as those seen in Jupiter’s atmosphere. The results include a study showing that equatorial westerly jets can be generated from decaying turbulence in a shallow-water system on a rotating sphere (Paper 6), and a study clarifying the mechanism of equatorial westerly jet generation from forced turbulence when the Newtonian cooling effect is taken into account (Paper 11). Furthermore, in the study of β-plane turbulence in the case where the approximation of the longwave limit is valid (paper 10), we found a new conserved quantity that is important for understanding the pattern formation.

I also conducted a study on the interaction between convection and zonal flows with an eye to planetary atmospheres, and found that there is a positive feedback relationship between convection and zonal shear in rotating systems, and that convection can enhance zonal shear (Paper 8).

Research on flow stability

I have been conducting theoretical and numerical studies on the stability of zonal flows, starting from the study on the stability of zonal flows as a model of the polar vortex in the winter stratosphere, which clarified the tracer transport associated with unstable waves (Papers 1 and 2). As an achievement for the nonlinear stability of zonal flow in two-dimensional incompressible fluid, I not only proposed a new method to calculate the upper bound of the amplitude of the growing unstable wave under the constraint of the conservation laws of the system (Paper 3), but also succeeded in proving analytically the identity of the two upper bounds found numerically (Paper 9).

Research on the development of numerical methods

Numerical computations of the fluid equations on a sphere are required for the research of items 1 and 2 above. The spectral harmonic transform method is used for these numerical computations, but in order to perform high-resolution computations efficiently on a computer, various tunings of the algorithm and memory saving are required. Therefore, in order to provide not only my own research but also the use of other researchers, I have made my implementation of the spectral harmonic transform method available to the public as the numerical library ISPACK, and have published some of our innovations in papers. In particular, I have succeeded in deriving a new efficient recurrence formula for the associated Legendre functions (Paper 13), which is also used in the spherical harmonic transform library developed by other authors. The know-how accumulated in the development of the numerical library has been published as a textbook (Book 1).

In addition to numerical computations on the sphere, new proposals are made for numerical computations of the shallow water e quation in a disk domain (Papers 4 and 5) and the fluid equation in an infinite domain (Paper 7). Furthermore, a high-accuracy numerical method for nonlinear steady-state solutions of lee wave is proposed (Paper 12).

I have been developing and publishing a FORTRAN library for scientific computing ISPACK.

- K. Ishioka (2004): Introduction to spectral methods.
*University of Tokyo Press*, 232pp. (in Japanese)

K. Ishioka and S. Yoden (1994): Non-linear evolution of a barotropically unstable circumpolar vortex.

*Journal of the Meteorological Society of Japan*,**72**, 63–80.K. Ishioka and S. Yoden (1995): Nonlinear aspects of a barotropically unstable polar vortex: flow regimes and tracer transport.

*Journal of the Meteorological Society of Japan*,**73**, 201–212.K. Ishioka and S. Yoden (1996): Numerical methods of estimating bounds on the non-linear saturation of barotropic instability.

*Journal of the Meteorological Society of Japan*,**74**, 167–174.K. Ishioka (2003): Spectral method for shallow-water equation on a disk — I. Basic formulation.

*Journal of Japan Society of Fluid Mechanics*,**22**, 345–358.K. Ishioka (2003): Spectral method for shallow-water equation on a disk — II. Numerical examples.

*Journal of Japan Society of Fluid Mechanics*,**22**, 429–441.Y. Kitamura and K. Ishioka (2007): Equatorial jets in decaying shallow-water turbulence on a rotating sphere,

*Journal of the Atmospheric Sciences*,**64**, 3340–3353.K. Ishioka (2008): A Spectral Method for Unbounded Domains and its Application to Wave Equations in Geophysical Fluid Dynamics.

*Proceedings of the IUTAM Symposium on Computational Physics and New Perspectives in Turbulence, Y. Kaneda(Ed.), Springer, IUTAM BOOKSERIES*,**4**, 291–296.N. Saito and K. Ishioka (2011): Interaction between thermal convection and mean flow in a rotating system with a tilted axis.

*Fluid Dynamics Research*,**43**, 065503.K. Ishioka (2013): A Proof for the Equivalence of Two Upper Bounds for the Growth of Disturbances from Barotropic Instability,

*Journal of the Meteorological Society of Japan*,**91**, 843–850.I. Saito and K. Ishioka (2013): Angular distribution of energy spectrum in two-dimensional β-plane turbulence in the long-wave limit.

*Physics of Fluids*,**25**, 076602.I. Saito and K. Ishioka (2015): Mechanism for the formation of equatorial superrotation in forced shallow-water turbulence with Newtonian cooling.

*Journal of the Atmospheric Sciences*,**72**, 1466–1483.S. Masuda and K. Ishioka (2015): A method to calculate steady lee-wave solutions with high-accuracy.

*SOLA*,**11**, 85–89.K. Ishioka (2018): A New Recurrence Formula for Efficient Computation of Spherical Harmonic Transform.

*Journal of the Meteorological Society of Japan*,**96**, 241–249.K. Ishioka, N. Yamamoto, and M. Fujita (2022): A Formulation of a Three-Dimensional Spectral Model for the Primitive Equations.

*Journal of the Meteorological Society of Japan*,**100**, 445–469.K. Ryono and K. Ishioka (2022): New numerical methods for calculating statistical equilibria of two-dimensional turbulent flows, strictly based on the Miller-Robert-Sommeria theory.

*Fluid Dynamics Research*,**54**, 055505.