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