An Alternative Approach to Nonhydrostatic Modeling,
Part I: Cold Bubble Test

Z.I. Janjic^{1)}, J.P. Gerrity, Jr.^{1)}
and S. Nickovic^{2)}

As
reported previously by the authors, a nonhydrostatic model has been developed
using an alternative approach. The
model uses the vertical coordinate based on hydrostatic pressure (mass), and
the model equations are consistent with those discussed by Laprise (1992). The basic idea is to split the system of
nonhydrostatic equations into two parts: (a) the part that corresponds
basically to the hydrostatic system, except for higher order corrections due to
the vertical acceleration and (b) the system of equations that allows
computation of the corrections appearing in the first system due to the
vertical acceleration. The procedure
does not require linearization or approximation of any kind.

The
nonhydrostatic model designed using this approach can be considered as a
natural, evolutionary extension of a hydrostatic model. The nonhydrostatic effects are introduced in
a transparent way through an add-on module.
This module can be turned on or off depending on resolution, so that the
same model can be run in the hydrostatic mode at lower resolution with no extra
cost. This feature appears attractive for models designed for a wide range of
horizontal resolutions, and in particular for unified global and regional
forecasting systems. The proposed
concept has been applied within a three-dimensional model with full physics and
appears to be computationally robust at all resolutions, and efficient for NWP
applications.

In
order to demonstrate the soundness of the formulation, the nonhydrostatic model
should be able to reproduce important nonhydrostatic motions at very high
resolutions even though such resolutions may not be affordable for NWP
applications in the near future. These
motions include small-scale buoyancy driven flows, as well as the disturbances
of uniform flow induced by small orographic obstacles. In order to study such problems, a
two-dimensional version of the model has been developed. This version is restricted to the vertical
plane and ignores the Earth rotation.

The
cold bubble test was designed following the set-up of Straka et al. (1993) who
used this set-up in order to produce a well converged solution that would serve
as a reference for solutions obtained using lower resolution nonhydrostatic
models. In a neutrally stratified
atmosphere with the potential temperature of 300_{}, an initial cold disturbance was introduced in the
temperature field of the form

_{}, _{},

where

_{} *m*, _{}*m*, _{} *m*, _{}*m*.

The integration domain extended 40 *km* in the *x* direction, and the free surface was located at 442 *hPa*, i.e., at about 6400 *m*.
The center of the initial disturbance was in the middle of the domain in
the *x* direction, i.e., 20 *km* away from either of the lateral
boundaries. As in the main tests in the
Straka et al. (1993) study, the horizontal resolution was 100 *m*, and the vertical resolution was 100 *m* on the average. The time step was 0.3 *s*, which was proportionally considerably longer than that used by
Straka et al. There was no divergence
damping or Rayleigh damping. However,
lateral diffusion was applied in both horizontal and vertical directions. The diffusion coefficient was somewhat
enhanced compared to that used in the Straka et al. (1993) experiments in order
to reach about the same amount of dissipation.
Namely, in contrast to the Straka et al. (1993) experiments, in the
tests reported here there was no time filtering at all.

The initial potential
temperature and the potential temperatures after 300 *s*, 600 *s* and 900 *s* in the right hand part of the
integration domain extending from the center to 19200 *m*, and from the surface to 4600 *m
*are shown in Fig. 1. The u
component of the wind and the w component of the wind after 900 *s* are shown in Fig. 2 in the same
region. The dashed contours indicate
negative values. Comparison of these
figures with the Straka et al. (1993) converged reference solution using the 25
*m* grid size reveals a remarkable,
both quantitative and qualitative similarity.
As can be seen from the results summarized in Straka et al. (1993), the
present model did equally well or better than other established models
specifically designed for studying processes on these scales.

_________________________________________________

^{1)} NCEP/EMC,
5200 Auth Rd., Camp Springs, MD 20746

^{2)} University of
Athens, Greece and ICoD, University of Malta, Valetta

e-mail: zavisa.janjic@noaa.gov

Fig. 1. The
cold bubble test: initial potential temperature and the potential temperatures
after 300 *s*, 600 *s* and 900 *s* in the right
hand part of the integration domain extending from the center to 19200 m, and
from the surface to 4600 m. The grid
size is _{}and _{}.

Fig. 2. The cold bubble test: the u component of
wind and the w component of wind after 900 *s*
in the right hand part of the integration domain extending from the center to
19200 m, and from the surface to 4600 m.
The grid size is _{}and _{}. The dashed contours
indicate negative values.

REFERENCES

Laprise, R., 1992: The Euler equations
of motion with hydrostatic pressure asan independent variable. *Mon. Wea. Rev.*__,120__, 197‑207.

Straka, J.M., R.B. Wilhelmson, L.J.
Wicker, J.R. Anderson and K.K. Droegemeier, 1993: Numerical solutions of a
non-linear density current: a benchmark solution and comparisons. *Intl.
J. Numerical Methods in Fluids*, **17**,
1-22.