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


Z.I. Janjic1), J.P. Gerrity, Jr.1) and S. Nickovic2)


                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




                                                           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




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.



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.