Objective: Comprehensive understanding of hurricane dynamics with a numerical model that explicitly includes the cloud-scale moist convection interacting with the hurricane environment.

Rationale: The power source of a tropical cyclone is moist convection in the form of clouds. While the cyclone's own circulation creates the mechanism for organizing and supporting the convection in the eye wall and rain bands, the large-scale environment also exerts controlling influences on the convection, especially in the initial stage of a cyclone and, often, at critical moments of intensification. It is advisable to eliminate the gross parameterization of the convection from numerical models, and to include an explicit representation of convective clouds, in order to understand the multiscale dynamics of cyclones in a realistic environment.

Method: Two categories of problems are involved in the construction of a numerical model for the present goal. One is the representation of various physical processes, and the other the design of numerical techniques and strategies. Among the problems in the first category, the representation of moist convection has been addressed by the radical formulation by Ooyama (1990, 1995) of thermodynamics, as was proposed and extended to include parameterized microphysics of precipitation. An adequate representation of the boundary-layer processes is under investigation. It was earlier decided that the model would cover the broad range of required horizontal scales by a sequence of multiply nested domains, and also that the semi-implicit method of time integration would be adopted to avoid excessively small time steps. Thus, the focus of efforts in the second category has been the development of accurate but robust numerical procedures for two-way nesting in this setting. The principle of the procedures that we have successfully derived on a simple theoretical basis will be explained below. It is general enough to be adopted by other models with similar needs.

Accomplishment: The nesting of two or more domains of different resolutions in a model is well known for the numerical noise it generates at interfaces, and the following observation may be distilled from those experiences in the literature: For any nested model with a hyperbolic system of prognostic equations, accurate and noise-free two-way nesting requires: a) the computational phase speed of any wave, whether advected or propagating, must not change across the interface, if the wave is transmissible, i.e., if the wavelength is resolvable on the both sides of the interface; and b) waves that can not be resolved on the other side are not transmissible and should not be allowed to reach the interface. The first requirement is difficult to satisfy by low-order finite difference schemes without heavy filtering. Higher-order schemes may satisfy it in the domain interior but become ambiguous near the interface. Spectral methods on harmonic bases are excellent if the phase speed comparison is between two independent domains, but provide no freedom for interconnecting them. Other spectral bases that achieve global orthogonalization with spatially nonuniform weights may allow optional boundary conditions but the nonuniformity makes them extremely undesirable for solving hyperbolic equations. Thus, in order to infuse both the flexibility of choosing boundary conditions and the spatial uniformity of representation with the phase-speed accuracy of a spectral method, there is a need for new spectral bases. In this respect, it has been found that the bases of local cubic-splines, defined at equally spaced nodes, provide a satisfactory spectral representation in a finite domain. The size of the nodal interval,

Key references:

DeMaria, M., S. D. Aberson, **K. V. Ooyama** and S. J. Lord,
1992: A nested spectral model for hurricane track forecasting. *Mon.
Wea. Rev.*, **120**, 1628-1643.

**Ooyama, K. V.**, 1987: Scale-controlled objective analysis.
*Mon. Wea. Rev.*, **115**, 2479-2506.

**Ooyama, K. V.**, 1990: A thermodynamic foundation for
modeling the moist atmosphere. *J. Atmos. Sci..*, **47**,
2580-2593.

**Ooyama, K. V.**, 1995: A thermodynamic foundation for
modeling the moist atmosphere. Part II: Tests of microphysics in the
formation of squall lines. *Preprints. 21st Conf. on Hurr. & Trop.
Meteorol.*, 219-221.

**Ooyama, K. V.**, 1997: The semi-implicit integration of a
nested spectral model and the result of tests in squall-line simulation.
*Preprints. 22nd Conf. on Hurr. & Trop. Meteorol.*, 531-532.

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*Last modified: 8/18/97*