Modeling the hurricane dynamics with explicit moist convection

Principal Investigator: K. V. Ooyama

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, delta-x, determines the spatial resolution. The basic premises and early applications of this method, called the Spectral Application of Finite-Element Representation (SAFER), were presented in Ooyama (1987) and DeMaria et al. (1992).

Cubic splines are continuously differentiable up to the second order, and if the finite time step, delta-t, is adequately small for computational stability, the calculated speed of either advected or propagating waves is very accurate for all wavelengths that are greater than 3 delta-x. The graininess of the representation affects those waves that are shorter than this limit, but they are removable by a low-pass filter of a sharp cutoff at the Nyquist wavelength. Such a filter, one with a 6th-order taper at a desired (or spatially variable) cutoff wavelength, may be built into the spectral transform by including a third-order derivative constraint in the definition of the transform.

If the nodal interval is doubled in the next coarser domain, the minimum wavelength of waves that are representable in the coarser resolution is about 6 times delta-x of the current domain. Since the wave speed does not change across the interface, those waves are also transmissible. Thus, the second requirement (b) for noise-free nesting is satisfied by gradually increasing the cutoff-wavelength of the built-in filter from the Nyquist wavelength of the current domain to that of the coarser domain in a transitional zone bordering the interface but within the current domain. If the wave speed had slowed in the coarser domain as it would with the finite difference method, the minimum transmissible wavelength would have to be much greater due to the foreshortening of shorter waves.

There is one problem that is peculiar to the nested spectral model. The conversion of a spatial field to the finite set of nodal amplitudes is by an integral transform. In the usual spectral model on a single domain, the integration is over the entire domain. In the nested model, however, any domain, except for the innermost, contains subdomains, where the spectra of signals extend to finer-scales that are presently not resolvable. An adequate sampling in the subdomains can eliminate aliasing, and the resulting spectra should be truncated at the limit of current resolution. This is correct in the sense of spectral transforms. In the inverse transform of the truncated spectra, however, false oscillations, known as Gibbs' phenomena, appear in the spatial fields and are imported back to the subdomains through the interface. The effects of this feedback may accumulate in time and destroy the validity of the entire model.

A clean solution to the above problem, which also proves to be most effective, is to remove the cause of the feedback by taking only a narrow border strip of the immediate subdomain into the integral transform on the current domain; the remaining interior of the subdomain is simply excluded. If the boundary strip is wider than c times delta-t, where c is the fastest speed of waves, the exclusion is physically justified for hyperbolic equations. If the time integration is explicit, the strip needs to be only as wide as one delta-x.

Thus, the spectral transform of spatial fields to amplitudes is made on a domain with an empty hole. At the interface with the superdomain, three inhomogeneous conditions are set to ensure that the field and its normal derivatives up to the second order are continuous across the interface. These conditions make the interface "invisible" to the waves that move in from the superdomain. At the inside interface with the hole, no boundary condition is imposed so that the transmissible waves from the subdomain strip will enter the domain unmodified. Since the procedure is recursively applied to every domain in the nesting sequence, communication at each time step occurs both ways across the interface.

The atmosphere transmits fast-moving acoustic waves, although they are believed to play insignificant roles in determining the course of meteorological events. If the model equations are integrated explicitly, however, the time step, delta-t, has to be excessively small in order to keep the acoustic waves computationally stable. For computational economy, we have chosen the semi-implicit method of time-integration to ameliorate the acoustic problem. In this method, the model equations are explicitly predicted as described above but with delta-t that is small enough only for the meteorological signals. The result is then adjusted to stabilize acoustic waves.

The required adjustments are calculated by solving a second-order elliptic equation with a forcing term that comprises the second-order time difference of the current and two previous steps of the prediction. In application to the nested model, however, the elliptic adjustment equation can not be solved in a domain with a hole, especially with no boundary condition around the hole, in the same way that the hyperbolic equations for meteorological signals are solved for flawless nesting. The conflicting requirements of the elliptic and hyperbolic equations has proved to be a false dilemma, since the acoustic adjustments need not be nested or connected between domains. Thus, it suffices to solve the elliptic equation independently for each domain without a geometrical hole, but by setting the forcing term to zero in the area of the subdomain, while the nesting procedures for meteorological signals remain unaltered.

The semi-inplicit, spectral nesting method, as described above, has been tested with a two-dimensional vertical model of the moist atmosphere for squall lines (Ooyama 1997). Experiments with a convective cell growing right on the domain interface, and squall lines propagating across the interface, have demonstrated that the method really works as theorized. There was no need for computational diffusion or other means of noise suppression in the experiments involving vigorous moist convection.
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