The ABC model: a non-hydrostatic toy model for use in convective-scale data assimilation investigations

Abstract

Abstract. In developing methods for convective-scale data assimilation (DA), it is necessary to consider the full range of motions governed by the compressible Navier-Stokes equations (including non-hydrostatic and ageostrophic flow). These equations describe motion on a wide range of timescales with non-linear coupling. For the purpose of developing new DA techniques that suit the convective-scale problem, it is helpful to use so-called "toy models" that are easy to run and contain the same types of motion as the full equation set. Such a model needs to permit hydrostatic and geostrophic balance at large scales but allow imbalance at small scales, and in particular, it needs to exhibit intermittent convection-like behaviour. Existing "toy models" are not always sufficient for investigating these issues.A simplified system of intermediate complexity derived from the Euler equations is presented, which supports dispersive gravity and acoustic modes. In this system, the separation of timescales can be greatly reduced by changing the physical parameters. Unlike in existing toy models, this allows the acoustic modes to be treated explicitly and hence inexpensively. In addition, the non-linear coupling induced by the equation of state is simplified. This means that the gravity and acoustic modes are less coupled than in conventional models. A vertical slice formulation is used which contains only dry dynamics. The model is shown to give physically reasonable results, and convective behaviour is generated by localised compressible effects. This model provides an affordable and flexible framework within which some of the complex issues of convective-scale DA can later be investigated. The model is called the "ABC model" after the three tunable parameters introduced: A (the pure gravity wave frequency), B (the modulation of the divergent term in the continuity equation), and C (defining the compressibility).

The “ABC-DA system” (v1.4): a variational data assimilation system for convective scale assimilation research with a study of the impact of a balance constraint

“…As reported in Petrie et al (2017) the continuous equations (1) have been discretised in time and space (the current implementation uses a 360 × 60 (horizontal × vertical) element grid with a gridbox size of 1500 × 250 m. Variables are stored on an Arakawa C grid in the horizontal and Charney-Phillips in the vertical (see Fig. 1 of Petrie et al (2017)), and periodic boundary conditions are imposed in the horizontal to avoid the need for a driving model to provide lateral boundary conditions. The integration scheme used is the split-explicit, forward-backward scheme of Cullen and Davies (1991) with a main timestep of ∆t = 4 s.…”

“…The operator ∇ −1 x is defined as ∇ −1 x δv = (∂/∂x) −2 ∂(δv)/∂x, which is based on application of the Helmholtz theorem (see e.g. Petrie et al (2017), Sect. 4.1).…”

“…4.2 for ABC-DA). Since GB becomes a less important constraint at smaller scales (Berre, 2000;Bannister et al, 2011;Petrie et al, 2017), the research question arises concerning whether using such a constraint in the B-model is necessary, or indeed harmful in convective-scale DA. fields via a linear regression operator, trained from sample data.…”

The ABC-DA system (v1.4): a variational data assimilation system for convective-scale assimilation research with a study of the impact of a balance constraint

“…The ABC model comprises a set of simplified partial differential equations for a two-dimensional spatial grid (x and z) plus time (t), which are based on the Euler equations. This section summarises the ABC model, and the reader is directed to Petrie et al (2017) for the details. The model equations are as follows:…”

“…Studying convective-scale DA in operational systems is burdened severely by the cost and complexity of these systems. The DA system described in this paper has been designed in the same spirit as that of the convective-scale toy model (the ABC model; Petrie et al, 2017), i.e. with an emphasis on low cost and simplicity.…”