The non-hydrostatic model ATHAM is formulated in three dimensions. In addition, two different two-dimensional versions are implemented: one in cartesian and one in cylindrical coordinates. An implicit time stepping scheme is used, the conservation of mass and momentum is explicitly guaranteed by applying the flux form for the equations of motion and continuity of the tracers.
The initialisation of ATHAM is done by prescribing vertical profiles for horizontal wind, temperature, relative humidity and gaseous species in the background atmosphere. An arbitrary orography, e.g., shape and height of a volcano, can be defined at the lower boundary. The input of material at the source is specified by defining additional vertical velocities, temperature and composition of the ejecta at at least three active grid points. The simulation begins just after the decompression phase within the crater, small scale processes in the vicinity of the source in the hot temperature regime are not resolved in the concept of ATHAM. We focus on processes occurring in the plume in the range of about hundred meters to some tens of kilometers.
Volcanic ash particles occur typically in mass fractions greater than 95 % in volcanic plumes close to the vent. They can no longer be treated as passive tracers with the mean flow as in usual atmospheric models, but their impact on the dynamics of the system has to be considered. In the non-hydrostatic model ATHAM, particles are treated as active tracers: they can occur in any concentration and they can influence the dynamics of the system by contributing to the mixture density, pressure and heat content. In general, the description of such a multicomponent system of of gaseous, liquid and solid active tracers requires a set of dynamic and thermodynamic equations for each component including the interactions between them. However, the consideration of a higher number of grid points for investigating the mesoscale evolution of the plume, and the treatment of higher numbers of tracers is not possible with this concept.
Two main assumptions are applied in ATHAM to circumvent the problem of dealing with a very large equation system (Oberhuber et al., 98):
- Dynamic equilibrium:
we assume instantaneous exchange of momentum between particles and gas. All particles move with their terminal velocity relative to the mixture, which allows for the description of sedimentation.
- Thermal equilibrium:
tracers can act as a source of heat, but we expect the system to exchange heat instantaneously, the in situ temperature of the volume mean being identical to the individual in situ temperature of each tracer.
Both assumptions require that the particles modeled are small. The grain sizes of volcanic particles considered in ATHAM are well inside the range of validity as shown by (Herzog, 98; Oberhuber et al., 98). The only restriction resulting from our assumptions is that the time resolution of the model must be large compared to the time needed to achieve both equilibria. The dynamical features of a volcanic eruption we simulate set a lower limit for the time step, which should be in the order of a second. On the other hand, the great advantage of applying both dynamic and thermal equilibrium is the strong reduction in the number of prognostic equations. The dynamic behaviour of the gas-particle-mixture can now be described by five equations predicting the three momentum components, heat and pressure for the mixture. For each tracer one additional transport equation concerning its specific fall velocity is taken into account. Active tracers and dynamical variables are coupled by the bulk density and the heat capacity of the mixture via the equation of state. Further diagnostic equations are used to account for the interactions between the tracers and to close the system of dynamic equations. For a complete description of the model equations see Herzog, 98 and Oberhuber et al., 98.