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REVIEW the Quasigeostrophic Omega Equation: Reappraisal

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REVIEW the Quasigeostrophic Omega Equation: Reappraisal VOLUME 143 MONTHLY WEATHER REVIEW JANUARY 2015 REVIEW The Quasigeostrophic Omega Equation: Reappraisal, Refinements, and Relevance HUW C. DAVIES Institute for Atmospheric and Climate Science, ETH Zurich, Zurich, Switzerland (Manuscript received 29 March 2014, in final form 1 September 2014) ABSTRACT A two-component study is undertaken of the classical quasigeostrophic (QG) omega equation. First, a reappraisal is undertaken of extant formulations of the equation’s so-called forcing function. It pinpoints shortcomings of various formulations and prompts consideration of alternative forms. Particular consider- ation is given to the contribution of flow deformation to the forcing function, and to the role of the advection of the geostrophic flow by the thermal wind (the R vector). The latter is closely related to the Q vector, the horizontal component of the ageostrophic vorticity, and the forcing function itself. The reexamination pro- motes further examination of the physical interpretation and diagnostic use of the omega equation particu- larly for assessing richly structured subsynoptic flow features. Second, consideration is given to the dynamics associated with the equation and its more general utility. It is shown that the R vector is intrinsic to a quasigeostrophic cascade to finer-scaled flow, and that a fundamental feature of the QG omega equation—the in-phase relationship between cloud-diabatic heating and the at- tendant vertical velocity—has important potential ramifications for the assimilation of data in numerical weather prediction (NWP) models. Finally, it is shown that, in the context of considering NWP model output, mild generalizations of the quasigeostrophic R vector retain interpretative value for flow settings beyond geostrophy and warrant consideration when addressing some contemporary NWP challenges. 1. Introduction comparatively weak and 2) its direct computation from the horizontal divergence field is beset both by the need In essence the quasigeostrophic (QG) omega equa- to determine the comparatively small ageostrophic de- tion enables a qualitative indication to be gained of the parture of the flow away from geostrophy (Exner 1902; major synoptic-scale ascent regions from inspection of Jeffreys 1919) and by the lack of adequate observational the contemporaneous synoptic charts. Furthermore, it data (Margules 1904). permits a first-order quantitative estimation of the ver- The seeming impasse posed by these ramifications was tical velocity field to be computed given the specification elegantly circumvented by the flowering [see the reviews of the three-dimensional geostrophic flow field. One of Eliassen (1984), Phillips (1990),andBosart (1999)]of indication of its diagnostic value follows from noting the concept of quasigeostrophy, which hinges upon the that strong low-level ascent can promote cyclogenesis decomposition of the flow into a primary geostrophic and favor the occurrence of precipitation. component and a secondary ageostrophic component. The omega equation’s derivation was the culmination of Pivotal to its emergence and consolidation were two in- a half-century quest that had been plagued by the almost- gredients: first, the realization that there is an ageostrophic geostrophic character of synoptic-scale flow. Ramifications component that can be inferred from and ‘‘forced’’ directly of such a flow state are that 1) the vertical velocity field is by the geostrophic flow (Sutcliffe 1938, 1939, 1947)and second, the deduction that there is a quasigeostrophic flow whose evolution is determined by the geostrophic Corresponding author address: Huw C. Davies, Institute for At- mospheric and Climate Science, ETH Zürich, CHN, Universitätstrasse flow itself (Charney 1948; Eliassen 1949). 16, 8092 Zürich, Switzerland. An approximate form of the QG omega equation was E-mail: [email protected] set out by Bushby (1952), more formal derivations were DOI: 10.1175/MWR-D-14-00098.1 Ó 2015 American Meteorological Society 3 Unauthenticated | Downloaded 09/29/21 10:41 AM UTC 4 MONTHLY WEATHER REVIEW VOLUME 143 supplied by Fjortoft (1955) and Thompson (1961),and In light of the foregoing the objective of the present subsequent reformulations were developed by, among study is twofold. Following some preliminaries (section 2), others, Wiin-Nielsen (1959), Hoskins et al. (1978),and a synthesis and review is provided of the various extant Hoskins et al. (2003). formulations of the QG omega equation along with for- The equation has become a staple aid to diagnosing the mulating some alternative forms (section 3). Then con- nature, structure, and dynamics of synoptic-scale flow sideration is given to the relationship and current relevance phenomena. Early on it provided theoretically based suc- of QG dynamics to more general flow settings (section 4). cor for two empirically based nascent concepts of cyclo- genesis (see, e.g., Petterssen et al. 1962) that were linked 2. Preliminaries respectively to significant low-level thermal advection and a. Four formulations of the QG omega equation strong upper-level vorticity advection, and later these con- cepts become the basis for the erstwhile two-type classifi- The quasigeostrophic omega equation prescribes an cation of cyclogenesis (Petterssen and Smebye 1971). ageostrophic vertical velocity field that is inferable from Thereafter the equation has in its various formulations the geostrophic flow. To focus on the essence of this been used, for example, to diagnose the dynamics of in- equation, consider the form it takes for a Boussinesq fluid dividual events (see, e.g., Boyle and Bosart 1986; Rolfson in adiabatic flow above an f plane and possessing a uni- and Smith 1996; Pedder 1997; Bracken and Bosart 2000; form background mean-state stratification N.Inthislimit, Strahl and Smith 2001); supplement NWP guidance (e.g., and with height taken as the vertical coordinate, the Barnes 1985; Barnes and Colman 1994); distinguish be- equation for the vertical velocity w reduces to the form tween the dynamics of baroclinic synoptic-scale develop- 2=2 1 2›2 › 2 5 ment and subsynoptic across-frontal circulation patterns N Hw f0 w/ z F , (1) (e.g., Keyser et al. 1989; Martin 1999); explore the relative where f is the Coriolis parameter. For completeness amplitude and nature of the forcing at different elevations 0 and later use the full quasigeostrophic set of equations is (e.g., Trenberth 1978; Durran and Snellman 1987); infer provided in appendix A. From Eq. (1) it follows that in qualitatively the major regions of ascent (e.g., Hoskins and this limit-form w satisfies a second-order elliptic equa- Pedder 1980; Sanders and Hoskins 1990); examine the tion with constant coefficients, and as detailed below the contribution of diabatic heating to cyclogenesis (e.g., forcing term F is only a function of the geostrophic flow. Chang et al. 1984; Tsou et al. 1987; Strahl and Smith 2001); The expression for F can be cast in various forms that, study jet flow and frontogenesis at upper levels (Cammas although mathematically equivalent, are open to mark- and Ramond 1989; Lang and Martin 2010); establish edly different interpretation. The four most common classification schemes for cyclogenesis based upon upper- extant formulations of the term are versus lower-level forcing of vertical motion (Clough et al. 1996; Deveson et al. 2002; Gray and Dacre 2006); and 5 $ 2 =2 $ F f0›[(vG Á H)zG]/›z H[(vG Á H)b], (2a) associate distinctive potential vorticity features of the am- conventional formulation bient flow with the vertical velocity field (Hoskins et al. 5 f [2(›v /›z Á $ )z 1 DEF], (2b) 2003; Dixon et al. 2003; Plant et al. 2003). It has also often 0 G H G refined-Sutcliffe formulation been adopted as a diagnostic tool in oceanographic studies. 5 $ Nowadays it could be argued that recourse to the QG 2( H Á Q), and (2c) omega equation is obsolete. Current high-resolution re- Q-vector formulation search and numerical weather prediction (NWP) models 5 $ deliver the vertical velocity field as a model-output vari- f0›[(vG Á H)q]/›z able, but the resulting patterns are often somewhat noisy in 2 =2 1 2 2 2 $ [ H (f0/N) › /›z ]f(vG Á H)bg. (2d) appearance and not always readily interpretable. Again potential vorticity formulation NWP models now operate at subsynoptic spatial scales and the attendant flow dynamics is beyond the strict validity of Here the deformation (DEF) term in the refined Sutcliffe quasigeostrophy. Indeed a range of higher-order versions formulation takes the form (appendix B;seeWiin-Nielsen of the omega equation have been formulated to supplant 1959) 5 2 its use in diagnosing the secondary ageostrophic flow. DEF [D1(›D2/›z) D2(›D1/›z)], (3a) Also, many of today’s pressing dynamical and NWP where (D1, D2) are the deformation components of the challenges (such as predictability, data assimilation, and geostrophic flow in Cartesian coordinates, or it can be the representation/parameterization of near-grid-scale recast as processes) appear to be substantially unrelated to di- 2 agnosis undertaken with the QG omega equation. DEF 52D ›(2l)/›z, (3b) Unauthenticated | Downloaded 09/29/21 10:41 AM UTC JANUARY 2015 R E V I E W 5 where D is the total deformation of the geostrophic flow This formulation was devised to link the QG omega and l is the orientation of the dilatation axis. equation with the so-called potential vorticity (PV) per- Also the vector Q in Eq. (2c) can be written (Sanders spective (Hoskins et al. 1985). In the quasigeostrophic and Hoskins 1990) such that framework the PV perspective relates flow development to the evolution of the interior distribution of q and the 52$ u 3 › › Q j H *jfk vG/ sg, (3c) surface distribution of b. The link is emphasized by treat- ing the two terms of the forcing in Eq. (2d) separately, and with s denoting a local coordinate aligned along an then obtaining the solution to the omega equation as the isentrope such that in the Northern Hemisphere the cold sum of three components, w 5 w1 1 w2 1 w3, such that air located to the left.
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