A software package (OZO, Omega–Zwack–Okossi) was developed to diagnose the processes that affect vertical motions and geopotential height tendencies in weather systems simulated by the Weather Research and Forecasting (WRF) model. First, this software solves a generalised omega equation to calculate the vertical motions associated with different physical forcings: vorticity advection, thermal advection, friction, diabatic heating, and an imbalance term between vorticity and temperature tendencies. After this, the corresponding height tendencies are calculated with the Zwack–Okossi tendency equation. The resulting height tendency components thus contain both the direct effect from the forcing itself and the indirect effects (related to the vertical motion induced by the same forcing) of each physical mechanism. This approach has an advantage compared with previous studies with the Zwack–Okossi equation, in which vertical motions were used as an independent forcing but were typically found to compensate the effects of other forcings.

The software is currently tailored to use input from WRF simulations with Cartesian geometry. As an illustration, results for an idealised 10-day baroclinic wave simulation are presented. An excellent agreement is found between OZO and the direct WRF output for both the vertical motion and the height tendency fields. The individual vertical motion and height tendency components demonstrate the importance of both adiabatic and diabatic processes for the simulated cyclone. OZO is an open-source tool for both research and education, and the distribution of the software will be supported by the authors.

Today, high-resolution atmospheric reanalyses provide a three-dimensional
(3-D)
view on the evolution of synoptic-scale weather systems

Two variables that are of special interest in the study of synoptic-scale
weather systems are the geopotential height tendency and vertical motion

For the need of diagnostic tools, some software packages have been developed
to separate the contributions of each forcing to the vertical motion and
height tendency. DIONYSOS

Here, we introduce a software package Omega–Zwack–Okossi (OZO) that can be used for diagnosing
the contributions of different dynamical and physical processes to
atmospheric vertical motions and height tendencies. OZO calculates vertical
motion from a quasi-geostrophic and a generalised omega equation

In the following, we first introduce the equations solved by OZO: the two forms of the omega equation in Sect. 2.1 and the Zwack–Okossi height tendency equation in Sect. 2.2. The numerical techniques used in solving these equations are described in Sect. 3. We have tested the software using output from an idealised 25 km resolution WRF simulation described in Sect. 4. Section 5 provides some computational aspects of the software. The next two sections give an overview of the vertical motion (Sect. 6) and height tendency (Sect. 7) calculations for this simulation. Software limitations and plans for future development are presented in Sect. 8, and the conclusions are given in Sect. 9. Finally, information about the data and code availability is given in Sect. 10.

The omega equation is a diagnostic tool for estimating atmospheric vertical
motions and studying their physical and dynamical causes. Its well-known QG
form, obtained by combining the QG vorticity and thermodynamic equations,
infers vertical motion from geostrophic advection of absolute vorticity and
temperature

List of mathematical symbols.

Qualitatively, the QG omega equation indicates that cyclonic (anticyclonic)
vorticity advection increasing with height and a maximum of warm (cold)
advection should induce rising (sinking) motion in the atmosphere. However,
when deriving this equation, ageostrophic winds, diabatic heating, and
friction are neglected. In addition, hydrostatic stability is treated as a
constant and several terms in the vorticity equation are omitted. Although
Eq. (

The omega equation can be generalised by relaxing the QG approximations

Apart from the reorganisation of the terms in Eq. (

Because the operators

In the Zwack–Okossi method

The geostrophic vorticity tendency at level

For the calculations shown in this paper,

In Eq. (

Analogously with the vertical motion, the height tendency is divided in OZO
to the contributions of different physical and dynamical processes as

By substituting the vorticity equation and the thermodynamic equation into
Eq. (

Vorticity advection (

Thermal advection (

The ageostrophic vorticity tendency in Eq. (

All five terms also affect the vorticity and temperature tendencies indirectly via vertical motions, which are calculated for each of them separately with the generalised omega equation.

This results in the following new expressions:

The equation system used in this study has been adopted partly from

Earlier diagnostic tools have come close to our technique. The most similar
approach is probably used in the DIONYSOS

Following the Helmholtz theorem, the horizontal wind can be divided to
non-divergent (

OZO calculates the divergent part of the wind (

The non-divergent wind is obtained as the difference between

The first version of the OZO software package is tailored to use output from WRF simulations in idealised Cartesian geometry. The computational domain is periodic in the zonal direction, whereas symmetric boundary conditions are used at the northern and southern boundaries. Before the calculation, the WRF data need to be interpolated to pressure coordinates.

All of the right-hand side terms of the omega equation
(Eq.

Because the calculations are done in pressure coordinates, the lower boundary
of the domain does not correspond to the actual surface. To mitigate the
impact of this, vorticity and temperature advection, friction, diabatic
heating, and the ageostrophic vorticity tendency are all attenuated below the
actual surface by multiplying them by a factor varying from 0 to 1. The
multiplication factor at each level depends on how far down the level is
below the ground. For example, for a surface pressure of 950

The omega equation is solved using a multigrid algorithm

Having obtained the estimate

After each multigrid cycle, the maximum difference between the new and the
previous estimate of

OZO has four parameters for governing the multigrid algorithm, with the
following default values: the under relaxation coefficient (

In the Zwack–Okossi equation, geostrophic vorticity tendencies are converted
to geopotential height tendencies using Eq. (

In the omega equation, homogeneous boundary conditions (

WRF is a non-hydrostatic model and can generate atmospheric simulations using
real data or idealised configurations

The simulation presented in Sects. 6–7 was run for 10 days with a 30 min
output interval, in a domain of 4000 km

After running the simulation, data were interpolated from model levels to 19
evenly spaced pressure levels (1000, 950, …, 100 hPa). The
interpolation was done with WRF utility

The model output data contained all the variables needed in solving the generalised omega equation and the Zwack–Okossi equation: temperature, wind components, geopotential height, surface pressure, and parameterised diabatic heating and friction components. Diabatic heating and friction in WRF included contributions from various physical processes, such as cumulus convection, boundary layer physics and microphysics. The physical tendencies are not in the default WRF output, and need to be added by modifying the WRF registry file.

To study the performance of OZO at a resolution that is more similar to that
used in several earlier diagnostic studies of synoptic-scale dynamics

OZO can be run on a basic laptop with Linux environment, provided that standard NetCDF library, Intel's MKL and some Fortran compiler, preferably GNU's gfortran, are available. The source code of OZO is written in Fortran 90 standard and can be currently compiled only for a serial version.

The inversion of the left-hand side operator of the omega equation
(Eq.

The dependence of computing time on model resolution. Note that the numbers of grid points are per vertical level, whereas the computing times are per the whole 3-D domain.

Figure

The QG omega equation (Eq.

The majority of the differences between

The difference

A more comprehensive statistical evaluation of the calculated vertical
motions is given in Figs.

Correlation of the omega equation solutions with

RMS amplitudes of

Figure

Vertical motions induced by individual forcing terms at level
700 hPa at time 118 h.

Figure

In terms of the RMS amplitudes evaluated over the whole model area and the
last 8 days of the simulation, temperature advection makes the largest
contribution to the calculated

These results are partly consistent with similar calculations made for
observation-based analysis data

A further division of

In this subsection, the calculated total height tendencies are compared with height tendencies from the WRF simulation. The latter were estimated as central differences from the 30 min time series of the simulated geopotential heights.

As Fig.

Figure

Time mean

Figure

The correlation between the calculated and WRF height tendency is highest in
the upper troposphere, which is roughly 0.97. The correlation weakens
slightly closer the surface, but still exceeds 0.95. Thus, the calculated
height tendency is generally in very good agreement with the tendency
diagnosed directly from the WRF output. These correlations are comparable but
mostly slightly higher than those reported by

As Fig.

The contributions of the individual height tendency components at the
900

Friction (Fig.

The imbalance term (Fig.

Time series of individual height tendency components at the 900 hPa
level from the cyclone centre during the deepening period.

Figure

Height tendency associated with vorticity advection by the

Figure

In contrast to vorticity advection, thermal advection by divergent winds was found to cause a negligible height tendency (not shown).

The idealised baroclinic wave simulation provides an effective and widely used tool for studying cyclone dynamics in an easily controlled model environment. For this reason, we chose to begin the development work of OZO from a relatively simple Cartesian implementation. Nevertheless, the idealised Cartesian geometry obviously reduces the number of potential applications of OZO. We aim to extend OZO to more complex spherical grid applications in the future. However, in principle the use of OZO is possible with some limited-area real cases where spherical geometry has been used, if the data are regridded to Cartesian geometry afterwards. However, in this case, one must change the lateral boundary conditions of OZO since they are tailored for periodic model domain by default.

Another limiting factor in the current version of OZO is the weak scalability in the multigrid omega equation solver together with the lack of parallelisation of the source code. For high-resolution runs, this means significant slowing of the calculations (Table 2). To reduce the lengthy computing times, we are currently developing a parallel version of OZO, which uses a different solving method for the omega equation. We aim to release this parallel version by the end of the year 2017.

An issue associated with the physical interpretation of high-resolution
simulations should also be noted. In OZO, the vertical motion and height
tendency contributions of vorticity advection, thermal advection, friction,
and diabatic heating are calculated assuming geostrophic balance between the
vorticity and temperature tendencies, while the deviations from this balance
are attributed to the imbalance term. As the balance assumption is
increasingly violated at higher resolution, the imbalance term grows larger.
This complicates the interpretation of the results particularly when the
imbalance term opposes the other terms. Such compensation is indeed apparent
in our results for vertical motions in the lower troposphere, where the
contributions of thermal advection and diabatic heating are strongly opposed
by the imbalance term (Fig.

In this paper, a software package called OZO is introduced. OZO is a tool for investigating the physical and dynamical factors that affect atmospheric vertical motions and geopotential height tendencies, tailored for WRF simulations with idealised Cartesian geometry. As input to OZO, the output of the WRF model interpolated to evenly spaced pressure levels is required.

The generalised omega equation diagnoses the contributions of different
physical and dynamical processes to vertical motions: vorticity advection,
thermal advection, friction, diabatic heating, and imbalance between
temperature and vorticity tendencies. Then, analogously with the vertical
motion, the height tendencies associated with these forcings are calculated.
As an advance over traditional applications of the Zwack–Okossi equation

The calculated total vertical motions and height tendencies in the test case are generally in excellent agreement with the vertical motions and height tendencies diagnosed directly from the WRF simulations. The time-averaged correlation between the calculated and the WRF height tendency was 0.95–0.97 in the troposphere. For the vertical motion as well, a correlation of 0.95 was found in the mid-troposphere. Our analysis further illustrates the importance of both adiabatic and diabatic processes to atmospheric vertical motions and the development of the simulated cyclone.

The OZO software is applicable to different types of WRF simulations, as far as Cartesian geometry with periodic boundary conditions in the zonal direction is used. One example of potential applications are simulations with increased sea surface temperatures as the lower boundary condition. Combined with OZO, such simulations provide a simple framework for studying the changes in cyclone dynamics in a warmer climate.

The source code of OZO is freely available under MIT licence in GitHub
(

The authors declare that they have no conflict of interest.

We thank the Doctoral Programme of Atmospheric Sciences, University of Helsinki for financially supporting the work of M. Rantanen. The work of O. Stepanyuk was supported by the Maj and Thor Nessling foundation (project 201600119) and the work of O. Räty was supported by the Vilho, Yrjö, and Kalle Väisälä Foundation. V. A. Sinclair was supported by the Academy of Finland Centre of Excellence Program (grant 272041).Edited by: D. Ham Reviewed by: two anonymous referees