An Alternative
Approach to Nonhydrostatic Modeling, Part II: Warm Bubble Test

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

Continuing
the tests of soundness of our alternative nonhydrostatic model formulation, the
warm bubble test suggested by Drogemeier (1985) was performed. In a neutrally stratified atmosphere with
the potential temperature of 300_{}, an initial disturbance of the potential temperature was
introduced of the form

_{},

_{},

where

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

The
integration domain extended 20 *km* in
the *x* direction, and the free surface
was located at 135 *hPa*, i.e., at
about 13500 *m*. The center of the initial disturbance was in
the middle of the domain in the *x*
direction, i.e., 10 *km* away from
either of the lateral boundaries. The
horizontal resolution was 100 *m*, and
the vertical resolution was 100 *m* on
the average. The time step with this
spatial resolution was 0.3 *s* as
before.

The
cyclic boundary conditions were prescribed.
In addition to that, the Rayleigh damping was applied with the weight
proportional to

_{} *for* _{}.

Here, *d* is
the distance of the grid point from the point located at zero height in the
middle of the integration domain in the direction of the *x* axis. The maximum
distance _{} is defined as the
distance of the uppermost point at the left boundary from this point. In the semi-circular domain _{} the damping is not
applied, while in the rest of the integration domain it operates with the
intensity increasing with distance, as described by the formula. In the test _{} was set to 13200 *m*.
For the maximum distance, the weight reaches the maximum of 0.1. Note that with this arrangement, the
Rayleigh damping is applied mainly at the lateral boundaries.

The
divergence damping was not used, and there was no time filtering of the basic
variables. However, the diffusion
coefficients _{}along the *x* and _{} axes were, respectively,
0.002 and 0.002 for *u*, 0.002 and
0.002 for *T*, and 0.012 and 0.012 for *w*.

The
potential temperature deviation is presented after 360 *s*, 540 *s*, 720 *s* and 900 *s* in Fig. 1. The area shown
extends 16 *km* along the *x* axis, and from 0 *m* to 13200 *m* along the *z* axis.
The contour interval is 1_{}. The intensity of
the disturbance and the rate of its ascent generally agree with the results
reported elsewhere. The rate of ascent
also agrees well with that predicted in low resolution runs by Mendez-Nunez and
Carroll (1994).

______________________________________________

^{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. 4. The
potential temperature deviation after 360 *s*,
540 *s*, 720 *s* and 900 *s* in the warm
bubble test. The area shown extends 16 *km* along the *x* axis, and from 0 *m* to
13200 *m* along the *z* axis.
The contour interval is 1_{}.

REFERENCES

Mendez-Nunez, L.R. and J.J. Carroll,
1994: Application of the MacCormack scheme to atmospheric nonhydrostatic
models. *Mon. Wea. Rev.*, **122**,
984-1000.

Droegemeier, K.K., 1985: The numerical simulation of thunderstorm outflow dynamics. Ph.D. Disertation, University of Illinois at Urbana-Champaign, 695 pp.