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 H)zG]/›z H[(vG 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 $ )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 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 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 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. 2=2 1 2 2 2 5 $ Now we indicate how these various formulations can N Hw1 f0 › w1/›z f0›[(vG H)q]/›z be derived most directly from the full quasigeostrophic subject to w 5 0atz 5 0, (5a) set (appendix A). The conventional formulation (Fjortoft 1 1955; Thompson 1961) can be derived from taking the z w 52(1/N2)[(v $ )b], (5b) =2 2 G H derivative of Eq. (A5) and H of Eq. (A6) followed by using Eq. (A3) to yield the two-term expression of 2=2 1 2 2 2 5 N Hw3 f0 › w3/›z 0 Eq. (2a) for F. The two terms are related respectively to subject to w 52w at z 5 0. (5c) the vertical gradient of the vorticity advection and the 3 2 horizontal Laplacian of the thermal advection. It follows that w1 involves the interior q distribution and The refined Sutcliffe formulation (Fjortoft 1955; Wiin- w3 is a contribution arising from the advection of po- Nielsen 1959) also follows directly from Eqs. (A5) and tential temperature b at the surface, so that w1 and w3 (A6), but now noting from Eq. (A3) that are associated directly with the two ingredient of the PV

5 =2 perspective. In contrast w2 relates to advection on is- DG(f0›zG/›z)/Dt DG( Hb)/Dt entropic surfaces and acting alone would ensure a zero and yields the two-term expression of Eq. (2b) for F. thermal tendency (i.e., ›b/›t 5 0). The first term refers to the advection of vorticity zG by b. F forcing and the vertical velocity the ‘‘thermal wind’’ (›vG/›z) and it is the erstwhile tra- ditional Sutcliffe development term (Sutcliffe 1947), and Equation (1) indicates that w is a diagnostic variable the second term—the so-called deformation term—is as whose spatial distribution at a stipulated time can be defined earlier [Eqs. (3a) and (3b)]. determined by inverting the equation. This requires the The Q-vector formulation (Hoskins et al. 1978)is specification of the contemporaneous distribution of the obtained most directly by using the thermal wind rela- ‘‘forcing function’’ in the interior and the application of 5 / tionship, f0›vG/›z 5 k 3 $Hb, and noting following the bottom (wz50 0) and upper (wz/‘ 0) boundary Eliassen (1962) that conditions. Hence the formal solution of Eq. (1) pre- scribes the ageostrophic vertical velocity field w re- D (f ›v /›z)/Dt 5 D (k 3 $ b)/Dt. (4) G 0 G G H quired by the prevailing geostrophic flow to maintain geostrophy, and the w field can be obtained by solving Further manipulation yields a compact single-term the equation numerically. representation of the forcing [Eq. (2c)] in terms of the Equation (1) also provides an avenue for tackling the vector Q that is the product of the baroclinicity and the classical challenge of synoptic meteorology of inferring spatial variation in the strength and the direction of qualitatively the regions of major ascent and descent the geostrophic wind [Eq. (3c)]. from inspection of standard synoptic charts, and it is also Finally, the potential vorticity formulation (Hoskins a tool for diagnosing the linkage between observed co- et al. 2003) follows directly from recognizing that the herent features of the synoptic-scale flow and the ac- quasigeostrophic potential vorticity q is linked to the companying vertical velocity field. buoyancy b such that 1) SIMPLE QUALITATIVE AND QUANTITATIVE 5 =2 1 2 2 2 f0›q/›z [ H (f0/N) › /›z ]b. INFERENCES The in situ maintenance of this linkage with time yields A customary qualitative aid to the inference pro- a representation of the forcing [Eq. (2d)] that relates F cedure is to recognize that locally the following ap- to the vertical gradient of the advection of q by the proximate relationship holds: geostrophic flow and the three-dimensional Laplacian of =2 1 2 2 2 }2 the thermal advection. [ H (f0/N) › /›z ]w w

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FIG. 1. (a) A schematic of an ellipsoidal lens of uniform negative forcing (F 52F) with a vertical-to-lateral aspect ratio of (f0/N) located at a height h above the bottom boundary. (b) The pattern of vertical velocity associated with such a lens. Displayed is the nondimensional 2 2 vertical velocity w* 5w/(Fa /N )in(x*, z*) space with (x*, z*) 5 [x/a,(N/f0)/(z/a)], and the lens occupies the shaded region centered at a height of h* 5 2.5. The uneven contour intervals for w* are as labeled. (c) The two curves show the values of w* for a lens located at varying elevations h*. The long-dashed line displays w*z*5h* at the center of the lens, and the dotted line shows the value of w* directly beneath the lens at a fixed elevation z* 5 1.5d* corresponding typically to a height of 1.5 km. because w varies about a zero mean. On this basis a local This solution has been derived using the method of negative region of F forcing would be associated with images with an opposite sign anomaly placed at z 52h a positive value of w, and likewise positive forcing below the ground to ensure that w 5 0atz 5 0. For connotes descent. An alternative is to invoke the geo- consideration of the influence of a uniformly stratified metric interpretation of a Laplacian as the difference stratosphere located above the ellipse, see Clough et al. between the mean value of a scalar variable over a small (1996). To fulfill the continuity requirement of no net volume to that at its core. This implies that where w vertical mass flux across any given horizontal level the exhibits a local positive extremum then the F forcing isolated F forcing needs to be supplemented by a similar would have negative value, and likewise a negative ex- ellipsoidal region of uniform positive forcing located at tremum would connote positive forcing. the same elevation in the far field but far enough away A quantitative prescription of the linkage of F forcing for its contribution to be negligible in the neighborhood to the vertical velocity field w can be obtained for the of the negative forcing. simple setting shown in Fig. 1a. It comprises an ellip- The form of the solution [Eqs. (6a) and (6b)]in- soidal region of uniform negative forcing, F 52F, dicates that the ascent is a maximum within the core of located at a height h above the surface and occupying the forcing, decreases like (1/r)inthefarfieldbutwith 2 2 2 2 2 the region r , a with r 5 [x 1 y 1 (N/f0) (z 2 h) ]. a more rapid decrease in the vicinity of the surface. Note that the forcing region has a width-to-height aspect This is illustrated in Fig. 1b, which depicts the pattern ratio of ;100 for typical extratropical values of f0 and N. for a lens located in the lower troposphere at around The accompanying vertical velocity signature takes 2.5 km with the nondimensional vertical velocity, w* 5 the form (cf. Clough et al. 1996) w/(Fa2/N2), shown as a function of (x*, z*) 5 [x/a,

(N/f0)(z/a)]. The vertical velocity retains a value of 2 2 2 w* $ (3/4)w* within the forced region, but it reduces w 5 (1/2)(Fa /N )[1 2 (1/3)(r1/a) ] max to (1/10)w* at a lateral distance of approximately 2 2 2 # max (1/3)(Fa /N )(a/r2) for r a, (6a) x* 5 3.0. To further calibrate the nature of the response Fig. 1c 5 2 2 2 . w (1/3)(Fa /N )[a/r1 a/r2] for r a (6b) depicts two measures of the vertical velocity. The first measure is the dependence of w* at the center of the lens 2 2 2 2 1/2 with r6 5 [x 1 y 1 (N/f ) (z 2 (6)h) ] . 5 0 (i.e., w*z*5h*) to the height, h* (N/f0)(h/a), of the lens.

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FIG. 2. (left) Schematics of two dipole configurations characteristic of extratropical forcing and corresponding respectively (top) to a vertically aligned forcing pattern astride the tropopause at, say, (x*, y*, z*) 5 (0, 0, 9.0), and (bottom) to a horizontally aligned pattern in the lower troposphere centered at (x*, y*, z*) 5 (0, 0, 2.5). Each dipole element is assumed to be of uniform ellipsoidal forcing with strengths of F 5620F for the tropopause level and 6F for the lower troposphere configuration. (right) The resulting w* field at z* 51.5 in the (x*, z*) plane with the dash– dotted and the dotted profiles pertaining respectively to the vertical and horizontally aligned dipoles, respectively.

Above h* 5 4.0, corresponding typically to a height of 2) MORE REFINED CONFIGURATIONS approximately 4 km and for a 2-km-deep forcing region, w*z*5h* is comparatively unmodified by the presence of The contribution of a localized region of (say) nega- the bottom boundary. For a forcing region located in tive F forcing to the net forcing could be countered or the upper troposphere the maximum vertical velocity indeed negated by a region of strong positive forcing 2 2 approximates to wmax ’ (1/2)(Fa /N ). located in the near vicinity. This is noteworthy because The second measure depicted in Fig. 1c is the de- the atmosphere’s pattern of forcing often exhibits pendence of the vertical velocity at a fixed location in the dipole-like structures. lower troposphere (;1.5 km) directly beneath the forc- Schematics of two such common configurations are ing for different elevations of the lens. The dotted line shown in the left panels of Fig. 2. They comprise a ver- in Fig. 1c demonstrates that low-level ascent is favored tically aligned forcing pattern astride the tropopause at by low-level forcing whereas the influence of mid- to h* 5 9, and a horizontally aligned pattern in the lower upper-tropospheric forcing is substantially less effective. troposphere at h* 5 2.5. The main panel of Fig. 2 shows This result is germane to consideration of surface the pattern of w* at fixed nondimensional height of z* 5 cyclogenesis because ascent in the lower troposphere 1.5 for these two configurations. connotes an increase in the low-level vorticity [see The upper-level dipole reduces w*atz 5 1.5 beneath Eq. (A5)]. the dipole by a factor of ’2 compared to the response To relate the amplitude of the response to a prescribed for a single-signed region located aloft at h* 5 9 (note forcing consider a 2000-km-wide and troposphere- the different strengths assigned to the two configura- spanning forcing region characterized by a thermal tions and compare with Fig. 1c). Likewise the horizon- 2 2 wind of 5 m s 1 km 1 and a vorticity gradient along the tally aligned low-level dipole reduces w*atz 5 1.5 2 2 2 baroclinic zone of 2 3 10 5 s 1 (2000 km) 1.Forthe beneath one of the forced region by some 30% com- first term in the refined Sutcliffe formulation [Eq. (2b)] pared to an isolated single forcing region. It follows that 2 this equates to a value of less than 0.05 m s 1. the characteristic dipole structure of synoptic-scale

Unauthenticated | Downloaded 09/29/21 10:41 AM UTC 8 MONTHLY WEATHER REVIEW VOLUME 143 forcing can significantly modulate the amplitude and advection or strong upper-level vorticity advection. structure of the low-level ascent, and that the response However, it is noteworthy that it is the vertical de- of the low-level dipole is ’20 times stronger per unit of rivative of the vorticity advection that appears in the forcing than that of the upper-level configuration. QG forcing. In the foregoing we considered only ellipsoidal forcing elements with a large width-to-height aspect ratio. b. The refined Sutcliffe formulation However, forcing regions with a smaller aspect ratio can This formulation [Eqs. (2b), (3a), and (3b)] is Galilean be conceived as a vertical column of overlapping ellip- invariant, and moreover from a purely kinematic soidal elements, and the elements can be combined to standpoint is free from cancellation effects. The latter represent richer forcing patterns. deduction follows because geostrophic flow at any level comprises the two independent components of vorticity 3. Reappraisal and variants of the F formulation and deformation and the formulation’s two terms relate separately to these components. a. The conventional formulation 1) THE SUTCLIFFE TERM Interpretation of the terms in the conventional for- mulation for F [see Eq. (2a)] is beset by issues of The Sutcliffe forcing term, 2f0[(›vG/›z) $H]zG, re- mathematical uniqueness, physical meaningfulness, and lates to the vorticity gradient along the thermal wind, practical utility (Trenberth 1978; Hoskins et al. 1978; and inspection of a single chart portraying both the Durran and Snellman 1987; Walters 2001). Here we thermal and the pressure patterns generally provides briefly underscore these shortcomings. a ready indication of the regions of strong forcing. This is First, the expression is not Galilean-invariant in that it illustrated in Figs. 3a and 3b for the two archetypical can be modified to the form configurations of the so-called thermal steering and jet left-exit settings. For both configurations negative forc- 5 2 $ 2 =2 2 $ F f0›f[(vG c) H]zGg/›z H[(vG c) Hb] ing connoting ascent will be large along the baroclinic zone where it traverses away from the region of en- by the addition of an uniform translational velocity c.This hanced vorticity, and is consistent with respectively the leaves the net forcing unchanged while modulating the displacement of the low along the baroclinic zone and of relative amplitude of the two terms. For a given synoptic cyclogenesis in the left exit segment of the jet. Insight setting assigning a value for c corresponding to an estimate into the accompanying dynamics follows from consider- of the translational velocity of a prevailing synoptic system ing the poleward slope of the isentropic surfaces within could shed light on the system-relative dynamics. the baroclinic zone. The inferred patterns of ascent and Second, the two terms in this formulation contain descent are consistent with the flow associated with am- compensating contributions of the form bient vorticity moving on the isentropic surfaces. For the synoptic setting of an incipient frontal cyclone 6 f [(v $ )(›z /›z)], 0 G H G (see Fig. 6a) the Sutcliffe term is indicative of ascent along the warm front ahead of the cyclone and descent which can often be comparable in magnitude to that of on its rearward southwestern flank. the individual terms. This feature detracts from the physical meaningfulness of the latter terms. 2) THE DEFORMATION TERM Third, qualitative inference of the two forcing terms is cumbersome. For example, it is necessary to assess the The form of the deformation term, DEF 52D2›(2l)/›z, vorticity advection, (vG $H)zG, at adjacent elevations, appears to require assessing the difference in the angle and moreover a typical synoptic flow setting with a lo- of orientation of the deformation field’s dilatation axis at calized maximum of vorticity advection at the tropo- adjacent levels in the vertical, and the term is formally of pause equates to a zero contribution to the forcing at the same order as the Sutcliffe term itself. In practice it that level with an accompanying dipole structure above tends to be small in the midtroposphere where the ther- and below, and this in turn corresponds to a reduced mal gradient is often weak and almost coaligned with the effect in the far field (section 2). geostrophic flow (Wiin-Nielsen 1959), but it can be sig- By way of a counter to the foregoing shortcomings it nificant in amplitude in frontal and jet-stream regions is noteworthy that the two terms of this formulation where it usually exhibits a rich localized spatial structure appear to be linked to the two-category classification (see, e.g., Martin 1999). of surface cyclogenesis into frontal wave development An alternative form can be derived that enables a di- accompanied by either significant low-level thermal rect qualitative inference of the in situ DEF forcing

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FIG. 3. Illustrations of the Sutcliffe forcing term, 2f0[(›vG/›z) $H]zG, for the archetypical ‘‘thermal steering’’ and ‘‘jet exit’’ flow settings. Depicted are the pressure and thermal fields (solid and dashed contours respectively) and the accompanying vorticity pattern (shaded with the sign of the associated circulation as indicated). The black arrow denotes a region where the vorticity decreases along the thermal wind and is hence indicative of ascent according to the Sutcliffe term, and contrariwise the white arrow denotes vorticity increase along the thermal wind and thereby favoring descent. by inspecting a single surface (see appendix B). The key Sutcliffe term is zero. The deformation field D is such to the derivation is to recognize that the thermal wind, that l 5 0 (cf. Fig. 4a) whereas the deformation of the

T G 5 ›vG/›z, is such that T G 5 (1/f0)(k 3 $Hb), and thermal wind S is strongest astride the zone of enhanced hence the isentropes on a given surface can be viewed as baroclinicity with « ’1458 on the warm side and « ’ the streamfunction of the T field. 2458 on the cold side (cf. Fig. 4c). It follows that on the It then follows that the DEF term can be recast in the warm side of the zone g ’ 458, sin(2g) is maximized, and form (see appendix B) the DEF forcing favors ascent. Contrariwise on the cold side the forcing favors descent. DEF 52[(D)(S)] sin(2g), (7) The setting in Fig. 4f of a baroclinic zone traversing where, as before, (D, l) correspond to the total de- across a Rankine vortex configuration serves to em- formation and the orientation of the dilatation axes of phasize that, if the vorticity and its gradient are weak in the geostrophic flow, and now (S, «) correspond to the the outer regions of a cyclone, the contribution of the total deformation and dilatation angle of the ‘‘thermal Sutcliffe term is less effective, and that the prevailing wind’’ field, while g 5 (« 2 l) defines the relative ori- deformation and the orientation of the dilatation axes entation of the two dilatation axes. Using Eq. (7) a syn- favors descent to the east and ascent to the west of the optic chart depicting both the potential temperature cyclone. field and the pressure field allows a qualitative inference The settings in Figs. 4g and 4h relate to the character to be made of the locations of strong DEF by assessing of cold and warm fronts in different ambient flow envi- the regions of major D and S plus the local relative ronments. For these panels the flows comprise respec- orientation of their dilatation axes. tively of uniform negative (l 51458) and positive (l 5 This is illustrated in Fig. 4. First, the upper panels 2458) horizontal shear, while the thermal pattern cor- (Figs. 4a–d) depict prototypical deformation fields and responds in both settings to a highly idealized cold and the orientation of the attendant dilatation axes for re- warm frontal pattern. Again the Sutcliffe term itself is spectively: a pure deformation field, a uniform hori- zero. For Fig. 4g the relative orientation of the dilatation zontal shear and a jetlike profile, and a Rankine vortex axes favors a direct circulation at the cold front with comprising an inner core in solid body rotation and an warm air ascent and cold air descent, but with no DEF outer domain in irrotational flow. The lower panels forcing along the warm front. Contrariwise the presence (Figs. 4e–h) show the DEF forcing resulting from vari- of positive shear promotes a direct circulation along the ous simple combinations of these deformation signa- warm front. tures for the flow and the thermal wind. This dichotomous behavior is pertinent to consider- Figure 4e is for the canonical setting of pure de- ation of the relative behavior and movement of fronts formation field acting on a baroclinic zone that is aligned within a single cyclonic system [see the discussion in along the dilation axis, and for this configuration the Schultz and Vaughan (2011)] and is in harmony with the

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FIG. 4. Illustrations of the DEF 52[(D)(S)] sin(2g) forcing term of the refined Sutcliffe formulation. (a)–(d) The archetypical geo- strophic flow or thermal wind fields corresponding, respectively, to pure deformation, an uniform shear, a jetlike profile, and a Rankine vortex comprising a region of solid rotation surrounded by irrotational flow, and the double-arrowed dotted line signifies the orientation of the dilatation axis for each setting. Also shown are various combinations of these configurations: (e) the canonical deformation-induced frontogenesis setting of a baroclinic zone (colored contours) embedded in a pure deformation flow field (light black contours), (f) a baroclinic zone embedded within a Rankine vortex configuration, and (g),(h) an arched baroclinic zone embedded within, re- spectively, a uniform negative and positive horizontal shear. Again the double-arrowed dotted lines signify the orientation of the dilatation axes (black for the geostrophic flow, and red and blue for, respectively, the warm and cold flanks of baroclinic features). The positive and negative symbols indicate the regions of DEF forcing favorable respectively for ascent and descent. distinctive cyclonic and anticyclonic shear (Davies et al. also sheds light on the subtle nature of quasigeostrophic 1991) and LC1 and LC2 (Thorncroft et al. 1993) classi- flow. We comment in turn upon these aspects. fications of baroclinic development. (Likewise, in- 1) INFERENCE PROCEDURE spection of Fig. 6b hints that, for a frontal-wave cyclone setting, the DEF term favors frontogenesis along a re- In this formulation ascent will be favored by a distinct stricted segment of the cold front.) maximum of Q convergence (i.e., negative $H Q), and in the vicinity of such a region the Q vector will tend to c. The Q-vector formulation point toward the region of convergence and away from The Q-vector formulation of the forcing [Eq. (2c)] a divergent region. It follows from Eq. (3c) that on provides a compact single term expression, and avoids a synoptic chart a region of large Q will be characterized both non-Galilean and cancellation effects. It offers by a combination of strong baroclinicity and a marked a tractable approach to inferring the regions favorable change in the strength and/or direction of vG along the for ascent; although it has some intrinsic limitations, it isentropes, and furthermore the resulting Q vector will

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FIG. 5. Illustrations of the Q vector for three archetypical flow settings: (a),(b) the thermal steering and jet-exit configurations respectively (cf. Fig. 3), and (c) the canonical deformation-induced frontogenesis setting. Solid black arrows are indicators of the flow’s direction and strength, the dashed line refers to the change in the flow along the baroclinic zone (i.e., ›vG/›s), and the bold open arrow denotes the Q vector (obtained by rotating the vector of the wind change by 908 to the right). (d) The R vector (shaded arrow) for the frontogenesis setting [cf. (c)] and the implied ageostrophic circulation in a vertical plane across the front is as indicated. be aligned 908 to the right (in the Northern Hemisphere) 1 f z 52 (z2 2 D2) (8a) of the vector wind change. This deductive chain is il- 0 AG 2 G lustrated in Figs. 5a–c for three archetypical settings, and also Fig. 6c for the setting of an incipient frontal- and the relationship (Hoskins and Pedder 1980; Keyser wave cyclone. 1999) 4($ ^ Q) 5 f ›(z2 2 D2)/›z. (8b) 2) SOME LIMITATIONS 0 G In general the Q-vector formulation has met with A key point here is that it is only the irrotational com- widespread acceptance, but it has some potential short- ponent of Q that contributes to the F forcing. comings. First, the need to estimate the divergence term, Thus qualitative inference of the forcing by locating a

$H Q, is demanding if the flow has a rich subsynoptic dominant Q vector is conditional upon the rotational com- spatial structure. ponent (Qrot) not swamping or masking the signal of the Second, the two-dimensional Q vector decomposes irrotational component (Qirrot). Furthermore, recourse to into independent irrotational and rotational compo- evaluating $H Q more precisely as the net flux of Q from nents (see, e.g., Keyser 1999), that is, a given area might also be challenging if /Qirrot/ ; /Qrot/. Compare the computation of the horizontal divergence Q 5 Q 1 Q . irrot rot ($H v) of synoptic-scale flow when it is small compared to the vertical component of the vorticity. In effect the quasigeostrophic flow is accompanied by 3) QUASIGEOSTROPHIC DYNAMICS AND THE both an ageostrophic irrotational flow related to $H Q Q VECTOR and diagnosed by the QG omega equation, and an ageostrophic rotational flow related to $^Q. Subject to A starting point for the Q-vector formulation is two the caveat noted in appendix A, the latter is diagnosed equations derived directly from Eqs. (A4) and (A6), via Eq. (A8), that is, namely

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FIG. 6. Depictions of the various representations of the forcing for a synoptic setting characteristic of an incipient frontal cyclone (L). Shown in each panel are the pressure and thermal fields (solid and dashed contours respectively) on a lower tropospheric surface, and the remaining symbols are as adopted in Figs. 3–5.

52 2 2 3 Here we explore further features of the Q-vector for- DG(f0›vG/›z)/Dt f0R f0 [k (›vAG/›z)], (9a) mulation but now do so using the R vector. Note, how- 3 $ 51 2 2 3 $ ever, that these two vectors are formally closely linked by DG(k Hb)/Dt f0R N (k Hw), (9b) the relationship where R 5 (›v /›z $ )v , so that it relates to the G H G 52 3 advection of the geostrophic flow by the thermal wind. Q f0(k R), [As noted in appendix A, an additional term would ap- pear in Eq. (9a) unless the primacy of the pressure field and this linkage is further emphasized by recognizing in is asserted over the wind field.] harmony with the physical interpretation of R that, from

The subtle nature of QG dynamics [see Hoskins et al. Eq. (3c), Q } fj$Hu*j›vG/›sg. (1978)] is rendered transparent on examining the role of Aspects of the role and significance of the R vector R in Eqs. (9a) and (9b). Its contribution to the evolution and other closely related vectors have been noted by of the vertical shear f›vG/›zg is equal and opposite to its Davies-Jones (1991) in his generalization of the Q-vector influence upon the baroclinicity term fk 3 $Hbg.Thus approach to flow states beyond strict quasigeostrophy, the intrinsically geostrophic effect of R acts to destroy and by Steinacker (1992) in his application of the the thermal wind balance, and thereby geostrophy itself. Q-vector approach to study frontal dynamics. Also, the Therefore the remaining ageostrophic terms on the R vector equates in essence to the horizontal compo- right-hand side of Eq. (9) are required to concomitantly nent of the C vectors derived by Xu and Keyser (Xu restore that balance. In short, the maintenance of the 1992; Xu and Keyser 1993) in their extension of the primary geostrophic flow necessitates the existence of Q vector to three dimensions, and likewise to the curl a secondary ageostrophic flow. of the horizontal component of the P vector formulated

Unauthenticated | Downloaded 09/29/21 10:41 AM UTC JANUARY 2015 R E V I E W 13 by Kuo and Nuss (1995) in their consideration of the of Eqs. (9a) and (9b) yields the following equation for geopotential tendency equation. Note that the present the R vector: definition of the R vector differs from the R* used by 51 2 3 $ 2 2 3 Davies-Jones (1991, hereafter DJ91). 2f0R N [k Hw] f0 [k (›vAG/›z)]. (10a) In line with Hoskins et al. (1978) and the aforemen- tioned studies, equating the terms on the right-hand side It then follows that

52 2 3 2 2 $ 2f0R N fk [(f0/N) (›vAG/›z) ( Hw)]g 52 2 3 2 $ 5 2 N fk [(›vAG/›Z*) ( Hw)]g with Z* (N/f0) z 52 2h h 5 3 2 $ N 1 with 1 k [(›vAG/›Z*) ( Hw)]. (10b)

Hence, with a rescaled vertical coordinate (Z*), R is pro- contribution R dc to the line integral is only significant portional to the negative of the horizontal vorticity com- over a part of the contour. The challenge to infer the ponent (h) of the ageostrophic flow (vAG, w). The linkage dominant Q or R vector is equivalent with one requiring of the Q and R vectors to the vorticity of the ageostrophic the estimate of a ‘‘divergence’’ and the other a ‘‘curl.’’ flow is evident in embryonic form in Hoskins et al. (1978), The key feature of the Q-orR-vector formulation is and is rendered explicit in Steinacker (1992) and Xu (1992). that they condense the physics to consideration of the In applying Eq. (10b) directly to observed data again note spatial structure of a single, easily diagnosed vector, and the caveat discussed at the end of appendix A. moreover the dynamical attributes of the R vector forms In effect the amplitude and direction of the R vector is, a platform for some of the considerations in section 4. subject to the foregoing caveat, an indicator of the strength and sign of the ageostrophic circulation, such that the d. The potential vorticity formulation ageostrophic circulation is anticlockwise about the R The PV formulation [Eq. (2d)] is reminiscent to that vector. This is illustrated in Figs. 5d and 6d for some pre- of the conventional formulation (Eq. 2a), and it suffers viously considered settings. For example, for the canonical from the same shortcoming of non-Galilean invariance setting of deformation-induced frontogenesis (Fig. 5d) and compensating contributions (CC). The latter now the R vector points along the thermal wind and implies a take the form counterclockwise vorticity/circulation in the across-front 56 $ plane with warm air ascent and cold air descent. CC f0[(vG H)(›q/›z)]. (12) The formulation can nevertheless provide some in- 4) FORCING AND THE VECTOR F R sight into the dynamics of a single system by considering The formulation of the F forcing in terms of R follows (Hoskins et al. 2003) a system-relative framework and directly from applying the operator [k $^] to Eq. (10a) replacing Eq. (2d) by the set of Eqs. (5). Another sig- to yield nificant mitigating factor is that the potentially de- bilitating influence of the compensating contributions is 2=2 1 2 2 2 5 $ 3 N Hw f0 › w/›z 2f0[k ( R)]. (11a) nullified over much of the flow domain. This arises be- cause the atmosphere’s major interior ‘‘q features,’’ and The utility of this formulation becomes more apparent hence the regions with potentially strong contributions on using Stokes’s equation to note that ðð ð to the compensating contributions [Eq. (12)], are con- fined predominantly to highly localized regions near the [k ($ 3 R)] ds 5 R dc, (11b) S C jet stream and to the immediate vicinity of air parcels previously subject to strong diabatic heating. where S represents a specified horizontal surface area However, from a PV perspective the flow and thermal enclosed by the closed contour C. patterns can be reconstructed from the interior q and Thus an assessment of the net forcing over a selected surface b distributions, and the latter distributions carry horizontal area S can be obtained by examining the a far-field signal. Thus an alternative to the foregoing PV component of the R vector aligned along the contour C. formulation is to evaluate the net forcing at a given level A simple procedure to identify a region of strong forcing as the sum of contributions arising from the interior q is to locate the zone of strong baroclinicity and then distribution alone, the surface b distribution alone, and select a contour C in its neighborhood such that the their interaction. Such a procedure could be extended to

Unauthenticated | Downloaded 09/29/21 10:41 AM UTC 14 MONTHLY WEATHER REVIEW VOLUME 143 examine the contribution to the forcing of spatially co- An evolution equation for TWLs follows from in- herent interior q and surface b features that are deemed troducing the thermal wind relationship Eq. (A2) into to be of seminal dynamical significance. Eq. (9b) to yield For example, consider the incipient frontal-wave cy- D (T )/Dt 5 (T $ )v 2 (N2/f )fk 3 $ (w)g. clone setting depicted in Fig. 6. The trough and atten- G G G H G 0 H dant flow resemble the signature at this level of an (13a) isolated upper-level q anomaly while the baroclinic zone Note that R 5 (T $ )v , and hence R can distort the is a manifestation of a surface front. The interaction of G H G TWLs either modifying their strength by along-flow these two signals would then influence the forcing at the stretching and/or reorienting them by deformation (see, displayed level. e.g., Keyser et al. 1988). The remaining ageostrophic The far-field contributions of q and b features also term modifies T by the tilting effect of the vertical ve- carry ramifications for the interpretative value of cy- G locity acting in the presence of a background vertical clogenesis classification schemes based upon the relative stratification. magnitude of the upper- and lower-level F forcing. Such At the surface w 5 0, and the equation reduces to schemes will not capture the interlevel forcing effects noted above. 51 $ DG(T G)/Dt (T G H)vG . (13b)

Hence at the surface the TWLs behave like ‘‘pseudo- 4. Relevance material lines’’ moving with the two-dimensional quasi- It was noted in the introduction that the contemporary geostrophic flow in the absence of frictional and diabatic relevance of the QG omega equation can be called into effects. An apposite analogy is with the reorientation and question on several accounts. Current research and deformation of vortex lines in a homogeneous three- NWP models deliver the vertical velocity field as dimensional flow (von Helmholtz 1858). From a similar a model output variable. High-resolution numerical standpoint and in the context of examining balanced flow models simulate subsynoptic flow features whose dy- DJ91 considered the evolution equation for the vertical namics, although often balanced, is beyond the strict shear of the horizontal wind. validity of quasigeostrophy. Moreover the dynamics of Here the dynamics of TWLs is intrinsic to so-called the QG omega equation appears unrelated to many of two-dimensional surface-geostrophic dynamics (see Held today’s pressing dynamical and NWP challenges. et al. 1995; Smith and Bernard 2013) and can produce In the following we address and at seek to at least a cascade to finer-scaled flow. An illustration of this partially alleviate some of the foregoing concerns. To physical scale cascade is provided in Fig. 7. The initial this end consider sequentially pertinent aspects of the state, in harmony with the so-called surface geostrophy, dynamics intrinsic to quasigeostrophic flow, diabatically comprises a surface warm band of b that decays with induced flow, and balanced flow. height accompanied by zero interior quasigeostrophic potential vorticity q. This configuration (see Fig. 7a)was a. Further comments on quasigeostrophy studied by Schär and Davies (1990) and is unstable. They There are two fundamental features of quasigeostrophy examined the growth rate of linear normal mode per- that bear directly upon its relevance to subsynoptic turbations and also numerically simulated its initial flow. nonlinear evolution. In the latter phase the pattern evolves to form a train of localized warm-cored vortices 1) CASCADE TO FINER-SCALED FLOW FEATURES that are linked together by scale-reduced warm bands Quasigeostrophic dynamics can itself induce finer- (see Figs. 7b,c). These new bands are themselves un- scaled flow systems and thereby contribute to the de- stable and breakdown (see Fig. 7d, and Fehlmann 1997) velopment of, and possibly account for a part of, the to form finer-scaled warm-cored vortices. panoply of the atmosphere’s subsynoptic flow features. The salient inference here is that quasigeostrophic This can be demonstrated by introducing the concept dynamics can contribute to the development of sub- of ‘‘thermal wind lines’’ (TWLs), defined as being ev- synoptic flow features, and intrinsic to this scale re- erywhere tangential to the local orientation of the duction is the role of the R vector, which is itself central thermal wind, T G 5 ›vG/›z. Hence the TWLs at any to the forcing of the QG omega equation. given level are aligned with the isentropes, and their 2) AGEOSTROPHY AND SCALE CONTRACTION strength proportional to the local baroclinicity (in- versely proportional to the isentrope spacing), and this Ageostrophy plays a key role in the development of linkage is only valid in the quasigeostrophic limit. some subsynoptic-scale flow phenomena such as fronts.

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FIG. 7. The quasigeostrophic development of a dimensional surface warm band with zero interior potential vorticity. (a) This unstable configuration is perturbed by its most unstable normal mode. The subsequent panels capture (b),(c) the evolution of a train of localized warm-cored vortices linked together by scale-reduced warm bands, which themselves develop (d) finer-scaled warm-cored vortices.

For example, ageostrophic effects substantially accel- insight into the flow that might strictly be beyond the erate the lateral contraction and refine the vertical essence of quasigeostrophy itself. structure of a baroclinic zone embedded within a synoptic- scale deformation field (see, e.g., Hoskins 1971; Davies b. Cloud-diabatic effects and Muller 1988; Reeder and Keyser 1988). However, The QG omega equation can serve as a test bed for this rapid frontal development is not captured ade- examining issues related to balanced flow. Consider- quately by the QG prediction Eq. (A7). Notwithstand- ation of the quasigeostrophic response to cloud diabatic ing diagnostic application of the QG omega equation at effects will serve to highlight a significant difference any given time during the scale-contraction process will between balanced and unbalanced flows. nevertheless, given pertinent information on the pre- Diabatic effects are customarily incorporated into the vailing flow configuration, deliver an estimate of the key QG omega equation by including an additional forcing ageostrophic flow component. Indeed for the particular term so that Eq. (1) becomes setting considered by Davies and Muller (1988) the sur- 2=2 1 2 2 2 5 1 =2 face ageostrophic flow component of the QG omega N Hw f0 › w/›z F HH, (14) equation corresponds to that obtained with the semi- geostrophic equations. where H isameasureofthediabaticheatingratewith In effect, application of the QG omega equation at aunitofH corresponding to approximately 3.0 3 2 a given instant can deliver useful information about and 10 2 degrees heating per day. The equation remains

Unauthenticated | Downloaded 09/29/21 10:41 AM UTC 16 MONTHLY WEATHER REVIEW VOLUME 143 self-contained provided H can be specified externally or in terms of geostrophic flow variables. =2 The H form of the diabatic forcing implies that the w field might not to be necessarily positive within the diabatic region itself (see Krishnamurti 1968; Hoskins et al. 2003). Here we explore and quantify this effect by considering the vertical velocity field associated with a localized positive lenticular ellipsoidal distribution of H of the form

H 5 Hf1 2 [(r/a)2 1 (z/d)2]g2 within the ellipsoid [(r/a)2 1(z/d)2] 51, (15a)

H 5 0 elsewhere. (15b)

Here r is the radial distance in cylindrical coordinates, and as before the horizontal-to-vertical aspect ratio of this forcing element is defined to be such that a 5

(N/f0)d. It follows from Eqs. (14) that the actual diabatic forcing pattern is given by

=2 52 2 2 2 2 2 HH (8H/a )[1 2(r/a) (z/d) ] (16) and the region of positive forcing is confined to within a narrower ellipsoid, FIG. 8. Nondimensional vertical velocity field, w* 5 w/H, asso- 2 2 2 [2(r/a) 1 (z /d) ] 5 1, ciated with a localized region of positive diabatic heating confined

within an ellipsoid with a vertical-to-lateral aspect ratio (f0/N)inan that is encased within an even narrower sheath of neg- otherwise unbounded domain. The response is shown in the non- ative forcing (cf. Hoskins et al. 2003). dimensional (r*, z*) plane, where r* is the radial distance, and the irregular contour intervals are as shown. Setting F 5 0 in Eq. (14), adopting a rescaled vertical coordinate, z* 5 (N/f0)z, and writing Eq. (14) in spher- ical polar coordinates (R, u, x), we have and w/H* 52(8/105)(a/R)3(1 2 3 cos2u) for R . a. (18b) =2w 52H*[1 2 2(R/a)2 1 (R/a)2 cos2u], (17) This pattern of vertical velocity corresponds to ascent 5 2 where H* 8H/(Na) . in a band narrower than the original diabatic region with Recalling that the Laplacian operator with longitudi- descent elsewhere (Fig. 8). The vertical velocity attains nal symmetry takes the form its maximum value wmax at the core of the diabatic re- 5 3 2 2 2 gion (i.e., at R 0) with a dimensional value of 2.0 = w 5 (1/R )f›/›R[R ›w/›R] 2 2 2 2 10 2 ms 1 for N 5 10 2 s 1 and a 10-K heating rate per 1 (1/sinu)›/›u[sinu›w/›u]g day. On the perimeter of the heating domain the vertical

velocity has decreased such that w ’ (1/4)wmax at the and solving Eq. (17) subject to continuity conditions at base (r, z) 5 [0, 2(f/N)a], and has reversed with w ’ 5 R a yields the following expression for the vertical 2(1/8)wmax on the lateral edge at (r, z) 5 [a, 0]. velocity: Note that the pattern for a cloud-diabatic region with a smaller width-to-height ratio would be equivalent to 5 2 2 2 w/H* (2/3)[1 (R/a) ] a vertical superposition of elements, and this would 2 (4/105)(R/a)2[7 2 5(R/a)2](1 2 3 cos2u) for strength the entire central shaft of ascending air. In the present context a key feature is that, in the R # a, quasigeostrophic limit, the vertical velocity accompanying (18a) diabatic heating is essentially in phase with the heating.

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This has dichotomous ramifications. For quasigeostrophic Likewise generalizations of the quasigeostrophic flow the phase match can be viewed as a mix of a positive omega equation have been formulated to diagnose feedback between the diabatic forcing and the response, the secondary ageostrophic flow component accompa- or a constraint upon the amplitude of the diabatic heating nying a primary balanced flow. These generalizations imposed by the moisture flux at cloud base attributable to fall into three classes. One class relates explicitly to the ascent forced by an ambient flow. An assessment of refined balanced systems such as semigeostrophic and the relative efficacy of these two effects could be obtained geostrophic momentum-like systems (Keyser et al. by applying the QG omega equation for a specified flow 1989; Keyser 1999; Pedder and Thorpe 1999), and a along with an estimate of the contemporaneous diabatic second class relates to the balance formulations fa- heating. vored in NWP (Ziemianski and Thorpe 2003). In the Another ramification is that for unbalanced flow a phase third class are equations derived directly from the match does not necessarily prevail between heating and primitive equations by extending either the procedure ascent. In the linear limit the vertical velocity response to followed to obtain the conventional formulation (see, stationary (Lüthi et al. 1989), steady (Davies 1987), or e.g., Krishnamurti 1968; Tarbell et al. 1981; Hirschberg pulsed (Lin and Smith 1986) imposed diabatic heating is in and Fritsch 1991; Pauley and Nieman 1992; Räisänen general out of phase with the heating. Likewise, the non- 1995) or the procedure for obtaining a generalized linear adjustment following a local instantaneous injection Q-vector formulation (e.g., DJ91; Viúdez et al. 1996). of heat in a resting ambient atmosphere yields a net de- This third class of equations contain time-tendency scent in the inner core of the initially warmed region with terms and are therefore pseudo, rather than strictly ascent above and below (Fanelli and Bannon 2005). diagnostic, omega equations. Nevertheless compari- This contrast in the heating–ascent relationship is son of w fields derived neglecting some terms, in- relevant in an NWP context. Longstanding challenges in cluding the time-tendency terms, with those delivered NWP are to deliver reliable quantitative forecasting of by the QG omega equation often show a resem- precipitation, and to develop robust and effective tech- blance beyond that expected by simple scaling argu- niques to assimilate/adjust model fields to account for ments (see e.g., Pauley and Nieman 1992; Räisänen misrepresented or unrepresented cloud diabatic heating. 1995). This success tallies with the experience of quali- The latter requires the amalgam of available information tative application of Q/R-vector approach to synoptic acquired from diverse data sources on the intensity, charts. movement, vertical profile, and location of cloud diabatic Here the objectives are to highlight the basis for this activity. One inference from the foregoing phase mis- effectiveness over and above the factors already noted in match is that an inadequate assimilation technique might section 4a. To this end mild generalizations of the qua- produce descent rather than the desired ascent in a region sigeostrophic R vector will be formulated, applied to in experiencing cloud-diabatic effects. the customary manner to examining NWP model out- More generally these ramifications highlight that put, and to examine their potential value in assessing the a higher form of balance must prevail between the dy- balanced versus unbalanced state of the output. The namics and thermodynamics in an active cloud-diabatic need for such an indicator has previously been high- region. To the extent that this form of hybrid balance lighted (e.g., Zhang et al. 2000). also pertains for diabatic processes at the grid/subgrid 1) R AND THE PSEUDO-OMEGA EQUATIONS scale, then it should also be taken into account in for- mulating stochastic representation schemes. We again consider the Boussinesq form of the primi- tive equations on an f plane, and follow the procedure of c. Balanced flow and the R-like vectors DJ91 and Viúdez et al. (1996) to obtain a pseudo-omega Some large-amplitude synoptic- and subsynoptic- equation. scale phenomena, although not strictly geostrophic, do The NWP model output fields for the horizontal wind exhibit a balanced character with the horizontal flow vF and the potential temperature u are nominally in- linked directly but nonlinearly to the prevailing pressure dependent, and neither need necessarily be represen- field. Such phenomena satisfy a state of balance beyond tative of a balanced model state. However, these fields quasigeostrophy and a range of predictive equations in- do provide independent measures for the vertical shear termediate between the QG and the primitive equations of the horizontal flow, namely a shear calculated di- have been developed for theoretical [e.g., geostrophic rectly, T F 5 ›(vF)/›z, and a ‘‘virtual’’ shear based upon momentum equations (Hoskins 1975), higher-order bal- the potential temperature and defined by the relation- ance, and Hamiltonian balance (Delahaies and Hydon ship, T u 5 [k 3 $Hb]/f0. The difference between these 2011)] and NWP (e.g., Bolin 1955) purposes. two measures,

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T 5 T 2 T R 5 R 5 R ›w ›z N2 AG F u , (19) F u , the / term is not present, and is prescribed as a function of height only. A further illu- defines and quantifies a departure of the model state minating example is a circularly symmetric baroclinic away from the thermal wind balance. vortex characterized by strong cyclostrophic balance so T Evolution equations for TF and T u can be derived that AG is appreciable. For this system Eq. (21) re- T 5 R 5 R 5 that are the analog of the quasigeostrophic Eqs. (9a) duces to DH[f0 AG]/Dt 0 because F u 0 and and (9b): w 5 0. In effect the prevailing magnitude of a flow’s departure from geostrophy, T AG, need not be an in- T 52 R 2 3 T 2 T D[f0 F ]/Dt f0 F -f0 (k AG) f0(›w/›z) F , dicator of imbalance. (20a) In DJ91 the slow evolution is assumed to be consistent with fluid parcels remaining in a state of thermal wind 2 T 52 R 2 3 $ 1 T equilibrium—that is, D[T AG]/Dt 5 0—and an omega D[f0 u]/Dt f0 u N (k Hw) f0(›w/›z) u . equation derived by applying the [k $^] operator to Eq. (20b) (21). Solution of the latter equation, given the horizontal Here D/Dt is the Lagrangian derivative following the velocity field and the thermal field, delivers a vertical model’s three-dimensional flow field, and N is now the velocity field forced by the prescribed (possibly un- T horizontal and vertically varying Brunt–Väisälä fre- balanced) fields. The influence of D[ AG]/Dt has been shown to be weak in some simple settings for which w 5 quency. Also RF 5 [T F $H]vF and Ru 5 [T u $H]vF, and these two horizontally aligned ‘‘R-like’’ vectors are 0(Davies-Jones 2009),andinthatstudytheaccompa- nying dynamical adjustment is characterized by inertial such that RF is a function of the model’s horizontal wind waves. The restriction to horizontal motion singles out field (vF), and Ru a function of the model’s thermal wind inertial waves as a mechanism for adjustment, whereas in (T u) and horizontal wind field (vF). Subtracting Eq. (20b) from Eq. (20a) and using Eq. (19) realized atmospheric flow the contribution of inertia buoyancy could be, and conceivably is, more significant. yields an evolution equation for T AG, Here the objective is to explore the value of a con- T 52 R 1 R 2 2h ventional R vector in examining the space–time output D[f0 AG]/Dt f0( F u) N 2 of NWP models. In particular emphasis is directed to the 2 T 1 T f0(›w/›z)( F u). (21) basis for its apparent effectiveness in diagnosing the major regions of ascent and to the insight it can shed on 2 Here h2 5 k 3 [(f0/N) (›vAG/›z) 2 $Hw] is akin to the balanced versus unbalanced state of the model state. the horizontal ageostrophic vorticity [cf. h1 defined in To this end consider two further formulations of Eq. (10)]. Equation (21) is equivalent to that derived in Eq. (21), DJ91, albeit using a different notation, and in that study T 1 T $ 52 R 2 2h 1 the first and the last terms on the right-hand side were DH[f0 AG]/Dt f0[ AG H]vF 2f0 u N 2 II taken together to define generalized R vectors. (22a) Equation (21) indicates that the Lagrangian rate of change (amplitude and orientation) of a fluid parcel’s and departure from thermal wind balance is determined by T 2 T $ 52 R 2 2h 1 the net of three terms: the sum of the two R-like vectors DH[f0 AG]/Dt f0[ AG H]vF 2f0 F N 2 II. involving only the horizontal velocity and thermal fields, (22b) a horizontal vorticity vector involving the vertical ve- locity field, and a third term involving both the hori- Here DH/Dt refers to the Lagrangian derivative following zontal flow variables and the vertical velocity. In the model’s horizontal flow (vF), [T AG $H]vF refers to principle each contribution could include strong geo- the horizontal stretching and deformation of T AG,and strophic and/or ageostrophic contributions because at 52 T T 2 T any given instant Eq. (21) encompasses both the ‘‘slow’’ II f0(›w/›z)( F u) f0w›( AG)/›z. evolution of a balanced flow component, and a possible rapid emission of inertia–gravity waves during a down- Thus Eqs. (22) involve only a single R-like vector and scale flow cascade plus the subsequent readjustment to all the terms that explicitly involve the vertical velocity a state of balance. are assembled on the right-hand side. Again applying One manifestation of the slow evolution is captured the [k $^] operator to Eqs. (22) would yield pseudo- by quasigeostrophy. In this limit Eq. (21) reduces to omega equations. Here we merely note that Eqs. (22) the diagnostic relationship Eq. (10a) because T AG 5 0, approximate to diagnostic relationships

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FIG. 9. The pattern of Qu vectors and the vertical velocity field (negative indicating ascent) on the 700-hPa level at 0000 UTC 1 Jan 2007. Fields are computed from the analysis fields of the ECMWF operational forecast model.

R ’2 2h 1 from the assimilation-cum-prediction cycle of the Eu- 2f0 u N 2 II, (23a) ropean Centre for Medium-Range Weather Forecasts T ’2 T $ if DH[f0 AG]/Dt f0[ AG H]vF , (23b) (ECMWF) operational forecast model. They allow an assessment to be made of the R or Q patterns of the R ’2 2h 1 and 2f0 F N 2 II, (24a) model fields. For example, Fig. 9 shows both the computed omega- T ’1 T $ Q if DH[f0 AG]/Dt f0[ AG H]vF . (24b) field and corresponding u-vector pattern. The consis- tency of the two fields over much of the domain points to Direct computations using space–time model output the not unsurprisingly balanced nature of the model will indicate the measure of applicability of Eq. (23b) or state, and concomitantly and more trenchantly to the (24b). If either Eq. (23b) or (24b) is applicable then we dominance of this form of forcing. There are discrep- recover a diagnostic omega equation, whereas inappli- ancies with some distinctive Qu-vector signals not ac- cability could be a helpful indicator of regions of un- companied by significant ascent, and these features can T balanced model flow. Note that if AG is predominantly be interpreted in terms of the remarks made in section 3 merely advected with and distorted by the horizontal regarding the properties of Q vectors. For example, the flow, then band of large Qu vectors that arches around the inner core of the incipient cyclone (L) is symptomatic of D [f T ]/Dt ’ f [T $ ]v H 0 AG 0 AG H F a strong rotational but nondivergent component to the Q field, and serves to emphasize the desirability of es- (cf. the analogy with material lines), and Eq. (24b) would timating ($ Q)or($ 3 R) to infer the forcing. be almost valid. This might prevail if the unbalanced Figure 10 pertains to a time 12 h ahead of the features flow comprises high-frequency, small-amplitude inertia– depicted in Fig. 9. The satellite image in Fig. 10a dem- buoyancy and buoyancy waves that pass through the onstrates that the event under consideration has the balanced flow pattern without stretching or deforming it classical structure of a rapidly developing cyclone. Fig- significantly. In a particular circumstance the selection of ure 10b shows the corresponding analyzed ECMWF Eq. (23a) or (24a) will depend upon the relative appro- thermal pattern and horizontal wind vectors on the priateness of Eq. (23b) or (24b). 700-hPa level at a given instant and Fig. 10c shows the

Qu vectors and omega fields. Qualitative inference of 2) MODEL OUTPUT AND R the regions of ascent using Fig 10b correspond relatively It is possible to infer Ru directly from the model well with the major features of the realized vertical ve- thermal and wind fields as set out in section 3. An al- locity field (Fig. 10c) including the strong across-front ternative is to consider a corresponding Q vector (Qu) circulations. The inferences also capture the movement also based upon the T u and vF fields. of the systems as is evident on comparing with the cor- Figures 9 and 10 pertain to an event of rapid cyclo- responding Qu vectors and omega fields of Fig. 9 that genesis over the western Atlantic, and were derived pertain for a time instant 12 h later.

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FIG. 10. Features of a rapidly developing cyclone over the western North Atlantic. (a) A Geostationary Operational Environmental Satellite (GOES) East satellite image (channel 1: visible-green to near IR) for 1500 UTC 31 Dec 2006 (i.e., 9 h ahead of the plots in Fig. 9). (b),(c) The 700-hPa surface, computed from the analysis fields of the ECMWF operational forecast model for 1200 UTC 31 Dec 2006: (b) depicts the analyzed isentropic pattern (colored) and the horizontal wind vector field, and (c) shows the Qu vectors and omega fields. The sat- ellite image is from the archives of NERC Satellite Receiving Station, Dundee University, Scotland (see http://www.sat.dundee.ac.uk/).

In effect adoption of the R-vector approach, based Again the relative amplitude and spatial structure of upon the generalization set out in Eq. (23a), appears to the T AG terms on the left-hand side of Eqs. (22a) and retain some interpretative value in a more general flow (22b) versus the R term can be readily evaluated and setting, and thereby lends credence to the theoretical assessed from model output. A significantly larger am- considerations underpinning the approach to the equa- plitude and spatially coherent region of the R term tion’s derivation. would be indicative of a locally balanced flow state,

Unauthenticated | Downloaded 09/29/21 10:41 AM UTC JANUARY 2015 R E V I E W 21 whereas a larger amplitude and richer or less coherent Acknowledgments. It is a pleasure to express my spatial structure to the sum of the T AG terms on the left- appreciation to Dan Keyser, Jonathan Martin, and hand side would be indicative of a nonbalanced flow Reinhold Steinacker for insightful discussions on as- contribution. pects of the QG omega equation, and to the reviewers Thus comparison of the terms can be instructive in for their fleet of helpful comments. My sincere thanks deciding whether it is useful to employ the corre- also to Rene Fehlmann for the material in Fig. 7 and to sponding pseudo-omega equation as a diagnostic tool Heini Wernli for preparing Figs. 9 and 10. and help diagnose the balanced or unbalanced nature of the local dynamics, and thereby concomitantly shed light on the effectiveness of assimilating data in specific APPENDIX A regions. Basic Relationships 5. Final comments The flow variables of the quasigeostrophic system are defined in terms of the departure (p*, r*, u*) of the The reappraisal of the classical quasigeostrophic pressure, density, and potential temperature away from omega equation undertaken in this study has served their horizontal mean values (p , r , u ). For the adia- several purposes. It has pinpointed possible shortcom- 0 0 0 batic flow of a Boussinesq fluid on an f plane, the geo- ings of using various extant formulations of the forcing strophic velocity, v , and the hydrostatic equation take to infer regions of ascent by inspection of standard G the respective forms synoptic charts, and prompted consideration of various alternative formulations. In particular it is shown that 5 3 $ 5 vG k Hc, and b f0›c/›z, (A1) the deformation term in the refined Sutcliffe formula- tion can be cast in a form that is illuminating from an where c 5 (p*/f0r0) is the geostrophic streamfunction, $ interpretative standpoint. Likewise the H Q formu- and b 5 (gu*/Q) is the buoyancy with Q referring to lation of the forcing can be recast in terms of the line a constant background value. integral of the R vector. Variants of the R vector have It follows that the thermal wind equation and the been considered previously, and it is closely related to equation for the vertical component of the relative the Q vector and defines, subject to a caveat regarding vorticity (zG) take the form the scale of QG flow, the horizontal component of the 5 3 $ ageostrophic vorticity accompanying a quasigeostrophic f0›vG/›z (k Hb), (A2) flow. 5 =2 A series of further considerations highlights the na- and f0›zG/›z Hb. (A3) ture of the QG omega equation. It is shown that the dynamics of the R vector can be intrinsic to a quasigeo- A power series expansion in terms of a Rossby num- strophic cascade to finer scaled physical flow systems. It ber provides (see Pedlosky 1964) the corresponding is demonstrated that in the QG setting there is an in- evolution equations for the geostrophic velocity, rela- phase relationship between forcing due to cloud- tive vorticity, and potential temperature: diabatic heating and the associated vertical velocity, 52 3 2 $ DGvG/Dt f0(k vAG) ( HpAG), (A4) and it is argued that this has implications for the as- similation of cloud-diabatic features in NWP models. 52 $ DGzG/Dt f0( H vAG), (A5) Finally it is shown theoretically and demonstrated em- pirically that a variant of the quasigeostrophic R vector 52 2 and DGb/Dt N w, (A6) retains some interpretative value in a more general flow $ 1 5 setting. In particular it provides a rationale for using an with ( H vAG) ›w/›z 0. extended form of the R vector to interpret model output of richly structured flow features, and a framework for Here N denotes a constant mean-state value of the assessing the balanced versus unbalanced state of NWP Brunt–Väisälä frequency, f0 is the Coriolis parameter model output. on an f plane, DG/Dt refers to the rate of change fol- Thus it is argued that the use of the QG omega lowing the horizontal geostrophic flow, (vAG, w) refer equation albeit in a refined form remains meaningful for to the ‘‘second-order’’ ageostrophic horizontal and interpretative purposes, and that the accompanying mild vertical velocities, and pAG denotes the leading-order generalization of the R vector is helpful when address- departure of the pressure field away from a base geo- ing some contemporary modeling challenges. strophic state.

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5 2 From the above it also follows that the evolution DEF [D1(›D2/›z) D2(›D1/›z)], (B1) equation for the quasigeostrophic potential vorticity q is given by where D1 5 (›yG/›x1›uG/›y)andD2 5 (›uG/›x 2›yG/›y) are the deformation components of the geostrophic flow in 5 5 =2 1 2 2 2 DGq/Dt 0 with q [ Hc (f0/N) › c/›z ], (A7) Cartesian coordinates. Equation (B1) can be reformulated in two suggestive the omega equation takes its conventional form, and ways. First, it can be cast in terms of the total defor- mation of the geostrophic flow, D 5 (D2 1 D2)1/2 and 1 1 2 f z 52 [z2 2 D2] 2 =2 c , (A8) the orientation (l) of its dilatation axis with respect to 0 AG 2 G H AG the x axis such that tan2l 5 (D1/D2). Noting that where zAG is the ageostrophic vorticity, and D is the total deformation of the geostrophic flow. A further 2 1 2 (D1 D2)›(D1/D2)/›z comment on the ageostrophic pressure field, pAG, ap- 52 2 1 2 2 2 pearing in Eq. (A4) and in Eq. (A8) is given below. (D1 D2)(1/D2)[D1(›D2/›z) D2(›D1/›z)] A compressed form of the QG equations is encap- 52[1 1 (D /D )2]DEF, sulated by the prognostic equation for q and the 1 2 diagnostic equation for w. The former provides a self- it follows that consistent equation for the evolution of ‘‘a’’ geostrophic 2 flow, and the latter determines the accompanying ageo- DEF 52D2[1 1 (tan2l)2] 1›(tan2l)/›z strophic vertical velocity field. In this study the latter 52 2 equation is to be used to infer the w field for a realized D ›(2l)/›z. (B2) flow state. Second, it can be cast in a form that incorporates the The appearance of the $HpAG term in Eq. (A4) in- dicates that to a leading order the selection of a given deformation of the thermal wind. The ‘‘thermal wind’’ T 5 T 5 3 $ pressure rather than a given velocity field to specify ›vG/›z is such that (1/f0)(k Hb), and hence T 5 the geostrophic flow remains open. In effect, given the two components of the thermal wind, (uG, yG), observational estimates of (perhaps incompatible) have deformation signatures given by pressure and horizontal velocity fields, an ambiguity either S 5 (›y /›x 1 ›u /›y) and arises in defining a geostrophic flow, and further dy- 1 G G 5 2 namical grounds are need to assert the primacy of one S2 (›uG/›x ›yG/›y) field relative to the other. For example, if the spatial scale of the flow is such that to the first order the flow 2 5 2 1 2 or equivalently a total deformation, S (S1 S2), and adjusts to the pressure field, then it would be appro- an orientation («) of its dilatation axis with respect to the priate to define the state using the pressure. In this limit x axis such that tan 2« 5 (S1/S2). pAG could then be neglected and the vertical compo- It then follows that DEF can be written successively as nent of the ageostrophic field, z , is determined by AG 5 2 Eq. (A8). DEF [D1S2 D2S1] (B3) In particular the primacy of the pressure field is not 5 2 assured for subsynoptic-scale flow features, and the [D sin(2l)S cos(2«) D cos(2l)S sin(2«)] observed pressure, temperature, and horizontal velocity 52fDSg sin(2g) (B4) fields might include ageostrophic signals. Thus there is a dynamical caveat to neglecting the $HpAG, and this is such that g 5 (« 2 l) is the relative orientation of the noted where necessary in the main body of the text. dilatation axes.

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