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OZO V.1.0: Software for Solving a Generalised Omega Equation and the Zwack–Okossi Height Tendency Equation Using WRF Model Output

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OZO V.1.0: Software for Solving a Generalised Omega Equation and the Zwack–Okossi Height Tendency Equation Using WRF Model Output Geosci. Model Dev., 10, 827–841, 2017 www.geosci-model-dev.net/10/827/2017/ doi:10.5194/gmd-10-827-2017 © Author(s) 2017. CC Attribution 3.0 License. OZO v.1.0: software for solving a generalised omega equation and the Zwack–Okossi height tendency equation using WRF model output Mika Rantanen1, Jouni Räisänen1, Juha Lento2, Oleg Stepanyuk1, Olle Räty1, Victoria A. Sinclair1, and Heikki Järvinen1 1Department of Physics, University Of Helsinki, Helsinki, Finland 2CSC – IT Center for Science, Espoo, Finland Correspondence to: Mika Rantanen (mika.p.rantanen@helsinki.fi) Received: 19 August 2016 – Discussion started: 29 September 2016 Revised: 3 February 2017 – Accepted: 6 February 2017 – Published: 21 February 2017 Abstract. A software package (OZO, Omega–Zwack– ucation, and the distribution of the software will be supported Okossi) was developed to diagnose the processes that af- by the authors. fect vertical motions and geopotential height tendencies in weather systems simulated by the Weather Research and Forecasting (WRF) model. First, this software solves a gen- eralised omega equation to calculate the vertical motions as- 1 Introduction sociated with different physical forcings: vorticity advection, thermal advection, friction, diabatic heating, and an imbal- Today, high-resolution atmospheric reanalyses provide a ance term between vorticity and temperature tendencies. Af- three-dimensional (3-D) view on the evolution of synoptic- ter this, the corresponding height tendencies are calculated scale weather systems (Dee et al., 2011; Rienecker et al., with the Zwack–Okossi tendency equation. The resulting 2011). On the other hand, simulations by atmospheric models height tendency components thus contain both the direct ef- allow for exploring the sensitivity of both real-world and ide- fect from the forcing itself and the indirect effects (related alised weather systems to factors such as the initial state (e.g. to the vertical motion induced by the same forcing) of each Leutbecher et al., 2002; Hoskins and Coutinho, 2005), lower physical mechanism. This approach has an advantage com- boundary conditions (e.g. Elguindi et al., 2005; Hirata et al., pared with previous studies with the Zwack–Okossi equa- 2016), and representation of sub-grid scale processes (e.g. tion, in which vertical motions were used as an independent Wernli et al., 2002; Liu et al., 2004; Beare, 2007). Neverthe- forcing but were typically found to compensate the effects of less, the complexity of atmospheric dynamics often makes other forcings. the physical interpretation of reanalysis data and model out- The software is currently tailored to use input from WRF put far from simple. Therefore, there is also a need for diag- simulations with Cartesian geometry. As an illustration, re- nostic methods that help to separate the effects of individual sults for an idealised 10-day baroclinic wave simulation are dynamical and physical processes on the structure and evo- presented. An excellent agreement is found between OZO lution of weather systems. and the direct WRF output for both the vertical motion and Two variables that are of special interest in the study of the height tendency fields. The individual vertical motion and synoptic-scale weather systems are the geopotential height height tendency components demonstrate the importance of tendency and vertical motion (Holton and Hakim, 2012). both adiabatic and diabatic processes for the simulated cy- Height tendencies are directly related to the movement clone. OZO is an open-source tool for both research and ed- and intensification or decay of low- and high-pressure sys- tems. Vertical motions affect atmospheric humidity, cloudi- ness, and precipitation. They also play a crucial role in Published by Copernicus Publications on behalf of the European Geosciences Union. 828 M. Rantanen et al.: OZO software atmospheric dynamics by inducing adiabatic temperature bining the QG vorticity and thermodynamic equations, infers changes, by generating cyclonic or anticyclonic vorticity, and vertical motion from geostrophic advection of absolute vor- by converting available potential energy to kinetic energy ticity and temperature (Holton, 1992): (Lorenz, 1955; Holton and Hakim, 2012). For the need of diagnostic tools, some software pack- LQG.!/ D FV (QG) C FT (QG); (1) ages have been developed to separate the contributions of each forcing to the vertical motion and height tendency. where DIONYSOS (Caron et al., 2006), a tool for analysing numer- @2! ically simulated weather systems, provides currently online L .!/ D σ .p/r2! C f 2 (2) QG 0 @p2 daily diagnostics for the output of numerical weather predic- tion models. RIP4 (Stoelinga, 2009) can calculate Q-vectors and the two right-hand side (RHS) terms are (Hoskins et al., 1978; Holton and Hakim, 2012) and a quasi- geostrophic (QG) vertical motion (Holton and Hakim, 2012) @ F D f V · r ζ C f ; (3) from Weather Forecast and Research (WRF) model output, V (QG) @p g g but the division of ! into contributions from various atmo- R F D r2 V · rT : (4) spheric processes is not possible in RIP4. Furthermore, many T (QG) p g research groups have developed tools for their own needs but do not have resources to distribute the software. (the notation is conventional, see Table1 for an explanation Here, we introduce a software package Omega–Zwack– of the symbols.) Okossi (OZO) that can be used for diagnosing the contribu- Qualitatively, the QG omega equation indicates that cy- tions of different dynamical and physical processes to atmo- clonic (anticyclonic) vorticity advection increasing with spheric vertical motions and height tendencies. OZO calcu- height and a maximum of warm (cold) advection should in- lates vertical motion from a quasi-geostrophic and a gener- duce rising (sinking) motion in the atmosphere. However, alised omega equation (Räisänen, 1995) while height tenden- when deriving this equation, ageostrophic winds, diabatic cies are calculated using the Zwack–Okossi tendency equa- heating, and friction are neglected. In addition, hydrostatic tion (Zwack and Okossi, 1986; Lupo and Smith, 1998). The stability is treated as a constant and several terms in the vor- current, first version of OZO has been tailored to use out- ticity equation are omitted. Although Eq. (1) often provides a put from the WRF model (Wang et al., 2007; Shamarock reasonable estimate of synoptic-scale vertical motions at ex- et al., 2008) simulations run with idealised Cartesian geome- tratropical latitudes (Räisänen, 1995), these approximations try. Due to the wide use of the WRF model, we expect OZO inevitably deteriorate the accuracy of the QG omega equation to be a useful open-source tool for both research and educa- solution. tion. The omega equation can be generalised by relaxing the QG In the following, we first introduce the equations solved approximations (e.g. Krishnamurti, 1968; Pauley and Nie- by OZO: the two forms of the omega equation in Sect. 2.1 man, 1992; Räisänen, 1995). Here we use the formulation and the Zwack–Okossi height tendency equation in Sect. 2.2. The numerical techniques used in solving these equations L.!/ D FV C FT C FV C FQ C FA; (5) are described in Sect. 3. We have tested the software us- ing output from an idealised 25 km resolution WRF simu- where lation described in Sect. 4. Section 5 provides some compu- @2! @2ζ tational aspects of the software. The next two sections give L.!/ Dr2(σ !/ C f (ζ C f / − f ! @p2 @p2 an overview of the vertical motion (Sect. 6) and height ten- @ @V dency (Sect. 7) calculations for this simulation. Software lim- C f k · × r! (6) itations and plans for future development are presented in @p @p Sect. 8, and the conclusions are given in Sect. 9. Finally, information about the data and code availability is given in is a generalised form of Eq. (2) and the RHS terms have the Sect. 10. expressions @ F D f V · r .ζ C f /; (7) V @p 2 Equations R 2 FT D r .V · rT /; (8) 2.1 Omega equation p @ FF D −f [k · .r × F /]; (9) The omega equation is a diagnostic tool for estimating atmo- @p spheric vertical motions and studying their physical and dy- R 2 FQ D − r Q; (10) namical causes. Its well-known QG form, obtained by com- cpp Geosci. Model Dev., 10, 827–841, 2017 www.geosci-model-dev.net/10/827/2017/ M. Rantanen et al.: OZO software 829 Table 1. List of mathematical symbols. −1 cp = 1004 J kg specific heat of dry air at constant volume f Coriolis parameter F forcing in the omega equation F friction force per unit mass g = 9.81 m s−2 gravitational acceleration k unit vector along the vertical axis L linear operator in the left-hand side of the omega equation p pressure Q diabatic heating rate per mass R = 287 J kg−1 gas constant of dry air D − @lnθ S T @p stability parameter in pressure coordinates t time T temperature V horizontal wind vector V g geostrophic wind vector α relaxation coefficient D − RT @θ σ pθ @p hydrostatic stability σ0 isobaric mean of hydrostatic stability ζ vertical component of relative vorticity ζg relative vorticity of geostrophic wind ζag relative vorticity of ageostrophic wind D dp ! dt isobaric vertical motion r horizontal nabla operator r2 horizontal laplacian operator @ @ζ R 2 @T FA D f C r : (11) from the geostrophic vorticity tendency. Neglecting the vari- @p @t p @t ation of the Coriolis parameter, Apart from the reorganisation of the terms in Eq. (6), this g ζ D r2Z (12) generalised omega equation is identical with the one used by g f Räisänen(1995). It follows directly from the isobaric prim- and hence itive equations, which assume hydrostatic balance but omit @Z f @ζg the other approximations in the QG theory.
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