2404 MONTHLY WEATHER REVIEW VOLUME 127
The Separate Roles of Geostrophic Vorticity and Deformation in the Midlatitude Occlusion Process
JONATHAN E. MARTIN Department of Atmospheric and Oceanic Sciences, University of WisconsinÐMadison, Madison, Wisconsin
(Manuscript received 19 June 1998, in ®nal form 30 October 1998)
ABSTRACT Separate vector expressions for the rate of change of direction of the potential temperature gradient vector
resulting from the geostrophic vorticity and geostrophic deformation, referred to as QVR and QDR, respectively, are derived. The evolution of the thermal structure and forcing for quasigeostrophic vertical motion in an occluded
cyclone are investigated by examining the distributions of QVR and QDR and their respective convergences. The dynamics of two common structural transformations observed in the evolution of occluded cyclones are revealed by consideration of these separate forcings. First, the tendency for the sea level pressure minimum to deepen northward and/or westward into the cold air west of the triple point is shown to be controlled by the
convergence of QVR, which is mathematically equivalent to thermal wind advection of geostrophic vorticity, a well-accepted mechanism for forcing of synoptic-scale vertical motion. Second, the lengthening of the occluded thermal ridge and surface occluded front are forced by the nonfrontogenetic geostrophic deformation, which rotates the cold frontal zone cyclonically while it rotates the warm frontal zone anticyclonically. The net result is a squeezing together of the two frontal zones along the thermal ridge and a lengthening of the occluded
thermal ridge. The associated convergence of QDR along the axis of the the thermal ridge also forces vertical motion on a frontal scale. This vertical motion accounts for the clouds and precipitation often observed to extend from the triple point westward to the sea level pressure minimum in the northwest quadrant of occluding cyclones.
1. Introduction expressions involved the along- and across-isentrope Extratropical cyclones are often accompanied by components of the Q-vector (Hoskins et al. 1978) and frontal baroclinic zones, usually a cold front and a warm so could be directly related to forcings for vertical mo- front. The thermal evolution of these cyclones repre- tion through the Q±G omega equation. sents a central component of their overall structural evo- In a recent study of the dynamics of occluded cy- lution. This thermal evolution involves changes in the clones, Martin (1999) employed a similar partitioning vigor of the individual frontal zones and in their ori- of the Q-vector into its along- and across-isentrope com- entation with respect to one anotherÐboth of which ponents (Qs and Qn, respectively). He showed that the occur throughout the cyclone life cycle. The dynamical predominant forcing for upward vertical motions in the processes that control these two components of the ther- occluded sector of midlatitude cyclones was associated mal evolution of cyclones are also responsible for pro- with convergence of Qs. It was further shown that the ١ implied by the convergent ducing secondary circulations to which the character- differential rotation of istic cloud and precipitation distribution in cyclones can ®eld of Qs was responsible for the production of the be accurately ascribed. characteristic occluded thermal ridge. Recent work by Keyser et al. (1988) and Keyser et In that study it was noted that Qs was the sum of al. (1992), hereafter referred to as K88 and K92, has contributions from both the geostrophic vorticity and examined the Lagrangian tendency of the potential tem- deformation (a point ®rst made by K88). These two perature gradient vector. K88 derived expressions for forcings were not, however, investigated separately. In the rates of change of both the magnitude and direction this paper, we extend our investigation of the rotational (i.e., Q ) component of Q by partitioning it into separate ١ and called them Fn and Fs, respectively. K92 s of demonstrated that the corresponding quasigeostrophic vector expressions representing the rates of change of ١ produced by the geostrophic vorticity direction of and deformation, respectively. The results of this anal- ysis shed new light on the Q±G dynamics of the occlu- Corresponding author address: Dr. Jonathan E. Martin, Depart- ment of Atmospheric and Oceanic Sciences, University of Wiscon- sion process by suggesting different, but complemen- sinÐMadison, 1225 W. Dayton St., Madison, WI 53706. tary, roles are played by the geostrophic vorticity and E-mail: [email protected] deformation in forcing the thermal evolution, and cloud
᭧ 1999 American Meteorological Society OCTOBER 1999 MARTIN 2405
V 2ץץ and precipitation distribution, of the postmature phase (١ (2a ´ midlatitude cyclone. ٌϩ22f 2 f g oog2 ഠ pץ pץThe analysis begins with a brief review of the various forms of the Q±G omega equation in section 2. We then on an f-plane except in frontal±jet streak regions where offer a derivation of the separate vorticity and defor- the neglected deformation term [term B in (2)] is usually mation contributions to Qs (the rotational component of large (Wiin-Nielsen 1959). In his examination of the the Q±G vector frontogenesis function) in section 3. importance of the deformation term throughout the life Examples of the separation of these two rotational com- cycle of a typical midlatitude cyclone, Martin (1998a) ponents in the three cyclones investigated by Martin has recently offered the more general statement that the (1999) will also be given there. The evolution of these deformation term is large in regions where ®rst-order separate forcings throughout the life cycle of one of discontinuities in temperature are coincident with re- these occluding cyclones will be given in section 4. A gions of nonzero ®rst derivatives in the geostrophic wind discussion of the occlusion process in light of this anal- ®eld. Such regions are not constrained to be frontal in ysis as well as the development of a dynamic conceptual nature; in fact, he showed that the thermal ridge com- model of the occlusion process are given in section 5. monly associated with the occluded quadrant of cy- Finally, in section 6 conclusions are offered along with clones is an example of a nonfrontal region in which suggested future directions. the deformation term is large in the midtroposphere. An alternative version of the Q±G omega equation on an f-plane was given by Hoskins et al. (1978) as 2. The quasigeostrophic omega equation 2ץ The traditional form of the Q±G omega equation is 22 (Q, (3 ´ ٌϩf o ϭϪ2١ p2ץgiven by (Holton 1992) where Q is given by 2ץ ٌϩ22f Vץ Vץ p2ץ o ˆ١ j ´ ١ i,Ã Ϫ ´ Q ϭ f ␥ Ϫ gg yץ xץo [] ץץ 2 1) c /c) , ١ ´ ϩ f )] Ϫٌ V)١ ´ ϭ f [V og g g with ␥ ϭ (R/fP )(P /P) p . Thus, the divergence of the p ooץp []ץ Q vector describes the complete forcing for the Q±G 2 2 2 2 2 2 p omega equation. Another important physical meaningץ/ץ y , g ϭ (1/ f o)ٌ , andץ/ ץx ϩץ/ ץwhere ٌ ϭ ϭϪRT/p. It is tempting to consider the two terms on of the Q vector is that it represents the rate of change ١ following the geostrophic ¯ow [i.e., Q ϭ the rhs of (1) as representations of separate physical of
.[yץ/ץ x ϩ Vgץ/ץ t ϩ Ugץ/ץ١, where d/dtg ϭ(processes; however, a convincing argument against such f o␥(d/dtg ١ a practice was made by Trenberth (1978) and Hoskins As such, Q describes changes in the magnitude of ,١. For this reason et al. (1978). Upon carrying out the derivatives on the as well as changes in the direction of rhs of (1), Trenberth (1978) found that some cancellation Q represents the Q±G analog of the vector frontogenesis existed between the two terms. By neglecting the so- function, F, introduced by K88. called ``deformation term'' (Wiin-Nielsen 1959) he con- Following a suggestion made by K88, K92 partitioned cluded that the rhs of (1) could be approximated by the Q vector into along- and across-isentrope compo- considering the advection of geostrophic absolute vor- nents and investigated the vertical motion forcings de- ticity by the column thermal wind, a result similar to scribed by each component in an idealized model sim- that produced by Sutcliffe (1947). The Trenberth (1978) ulation. Barnes and Colman (1993) and Kurz (1997) expression of (1) is given by offer recent examples of this partitioning in observed cases. We adopt a natural coordinate system in which ١ and sà is 90Њ counterclockwise nà is directed along 2ץ ٌϩ22f o 2 from nà (slightly different from that used by K88 and -p K92). In such a coordinate system the across- and alongץ isentrope components of Q (Q and Q , respectively) Vg n sץ ϩ f ) are given by)١ ´ϭ 2 f og pץ A ١ ١ ´ Q Q ϭϭQ nà (4a) |١||١|nn [] Vץ Vץ UץVggggץ (2) , ١ ´ Ϫ ١ ´ ϩ 2 f p andץx ץ pץy ץ[] o B (١ ١)(kˆ ϫ Q ´(kˆ ϫ Q ϭϭQ sÃ. (4b) |١[]||١|which is approximately equal to ss 2406 MONTHLY WEATHER REVIEW VOLUME 127
K92 found that bands of Q forcing distributed parallel half of the approximate Trenberth forcing for Q±G ver- to the baroclinic zones within their idealized cyclone tical motion (2a). Martin (1999) noted that this fact (physically reminiscent of frontal circulations) were en- accounts for the considerable similarity in distribution tirely accounted for by convergence of the across-is- that generally exists between regions of Trenberth forc- entrope component, Qn. Consistent with this inference, ing for upward motion and regions of Qs convergence the scalar function Qn is identically equal to the Q±G in the middle troposphere during the development and .١| ]. As a result, early mature stages of a typical midlatitude cyclone|(frontogenesis function [Fg ϭ (d/dtg Qn can be rewritten in terms of the kinematic properties Hoskins and Pedder (1980) showed that the f-plane ١| /2)[Eg version of the Trenberth forcing function (2a) could be|) of the geostrophic ¯ow as Qn ϭ Fg ϭ cos2], where Eg is the total resultant geostrophic de- written in a divergence form similar to that of the Q-vec- formation and  is the angle between the isentropes and tor forcing for Q±G omega. That form is given by the local axis of dilatation. This formulation reiterates Vgץ (QTR, (7 ´ ١ ϭϪ2١ ´a point implicit in Petterssen (1936); namely, that in 2 f og pץ purely geostrophic ¯ow the magnitude of the potential temperature gradient vector can be changed only by the ١). They did not attribute where QTR ϭ f o␥g(kˆ ϫ geostrophic deformation. physical signi®cance to Q Reference to (6), however, K92 also found that the forcing associated with the TR suggests that the vector QTR physically describes twice ١ produced by the geostrophic relative along-isentrope component, Qs, was distributed in a di- the rotation of pole on the scale of the synoptic disturbance in their vorticity, a point made by Martin (1999). model. They interpreted this fact to mean that conver- QTR for term A in (6) yields ´ Substituting Ϫ١ gence of Qs described the synoptic-scale forcing for Vgץ Q±G omega. They also showed that the function Qs can (Eg sin2). (8)١ ´ QTR ϩ f ´ Qs ϭϪ١ ´ Ϫ2١ pץ ١| d␣/dt, where ␣ is the orientation| be written as Qs ϭ angle of isentropes to the x axis (i.e., a latitude circle). Thus, the forcing for resulting from the rotation of Thus, Qs represents the rotational component of the Q±G ١ made by the geostrophic deformation is given by ١ being rotated vector frontogenesis function, with
Vץ .counterclockwise (clockwise) for positive (negative) Qs (E sin2) (9a)١ ´ Q ϭ f g ´ ١ Q ϩ ´ Ϫ2١ p gץ An extension of a derivation presented in K88 shows s TR that Qs can be expressed in terms of invariant kinematic properties of the wind ®eld as or ١| 1| Q ´ Q Ϫ Q ϵϪ2١ ´ Qsggϭ ( ϩ E sin2). (5) Ϫ2١ 2 s 2 TR DR
Vgץ -Thus, the rotational component of the Q±G vector front (Eg sin2), (9b)١ ´ ogenesis function consists of contributions from the geo- ϭ f pץ strophic relative vorticity (g) and the resultant geo- strophic deformation (Eg). In the next section we will where QDR is the vector representing the rate of change -١ produced by the geostrophic defor isolate these two contributions in separate vector ex- of direction of pressions, each of which contributes to forcing for Q±G mation. It can be shown (see appendixes A and B) that
١. the Cartesian form of QDR is given vertical motion and rotation of ץ 22ץ 1ץ ץ ␥ f ,(١ Q ϭ o E ϩ E Ϫ (kˆ ϫ 3. The vorticity and deformation contributions DR2 1 2 xץy ץy 2 ץ xץ[] |١| to Q s (10) a. Derivation -y) is the geostrophic stretchץ/Vgץx Ϫץ/Ugץ) where E1 ϭ y)istheץ/Ugץx ϩץ/Vgץ) Our partitioning of the rotational component of the ing deformation and E 2 ϭ Q±G vector frontogenesis function into its vorticity and geostrophic shearing deformation. We shall hereafter re- deformation contributions begins by considering the fer to the vorticity forcing contribution as QVR noting 1 convergence of Qs. Martin (1999) showed that that QVR ϭ 2QTR. Vץ Vץ (E sin2), (6)١ ´ ١ ϩ f ´Q ϭ f gg ´ Ϫ2١ p g b. Three examplesץp goץso AB In the study by Martin (1999) three different occluded where terms A and B represent the contributions to cyclones were considered and the partitioned Q-vector -١ made by the forcing for each was illustrated at some point after oc forcing resulting from the rotation of geostrophic relative vorticity and the geostrophic de- clusion [see Figs. 14, 15, and 16 of Martin (1999)]. In formation, respectively. Term A is precisely equal to each case, the vast majority of Q±G forcing for ascent OCTOBER 1999 MARTIN 2407
FIG. 1. The 18-h forecast from the UW-NMS valid at 0600 UTC 23 Oct 1996. (a) Solid lines are sea level isobars (labeled in hPa and contoured every 4 hPa) and dashed lines are 950-hPa potential temperatures (labeled in K and contoured every 2 K). Conventional frontal analyses indicate model-based positions of the surface fronts. (b) 600±900-hPa column-averaged Qs vectors and Qs convergence from an Ϫ1 Ϫ1 18-h forecast of the UW-NMS valid at 0600 UTC 23 Oct 1996. The Qs convergence is contoured and shaded in units of m kg s every Ϫ16 Ϫ1 Ϫ1 Ϫ16 Ϫ1 Ϫ1 5 ϫ 10 mkg s beginning at 5 ϫ 10 mkg s . (c) The 600±900-hPa column-averaged QVR vectors and QVR convergence from an 18-h forecast of the UW-NMS valid at 0600 UTC 23 Oct 1996. The QVR convergence is contoured and shaded as in Fig. 1b. Surface frontal analysis as in Fig. 1a. (d) The 600±900-hPa column-averaged QDR vectors and QDR convergence from an 18-h forecast of the UW-
NMS valid at 0600 23 Oct 1996. The QDR convergence is contoured and shaded as in Fig. 1b. Surface frontal analysis as in Fig. 1a. in the occluded sector of the cyclone was accounted for the model runs used to simulate the three cases illustrated
Qs). To illustrate the separation between the here, are described in detail in Martin (1999) and are not)´ by Ϫ2١ geostrophic vorticity and deformation contributions to repeated here.
Qs, we show the partitioned Qs forcing for those same At 0600 UTC 23 October 1996 a surface cyclone three cyclones in the ensuing ®gures. It is important to center was located over the Iowa±Wisconsin±Illinois note that the Q-vector forcings are calculated using col- border. The 950-hPa (Fig. 1a) along with the 950-hPa umn-averaged geostrophic wind and in the 500±900- absolute vorticity (not shown) were used to determine hPa layer (except in Fig. 1 where the 600±900 hPa layer the model forecast frontal positions. The surface oc- is used). The thermal ridge identi®ed by this column- cluded front1 was located in the ridge extending from average is nearly collocated with the lower-tropospheric thermal ridge used, in part, to determine the position of the surface-occluded front. All of the analyses in this paper employ output from numerical simulations of the 1 It is our opinion that occluded ``fronts'' do not represent boundaries between different air masses so much as boundaries between different selected cyclones performed using the University of Wis- baroclinic zones. Therefore, such boundaries are not ``fronts'' in the consin Nonhydrostatic Modeling System (UW-NMS). traditional (i.e., air mass) sense of the word. We retain use of the noun The model description, as well as the speci®cations for ``front'' as it is the historically accepted term for this feature. 2408 MONTHLY WEATHER REVIEW VOLUME 127
FIG. 2. (a) As for Fig. 1a except from a 12-h forecast by the UW-NMS valid at 1200 UTC 7 Nov 1996. (b) The 500±900-hPa column- averaged Qs vectors and Qs convergence from a 12-h forecast by the UW-NMS valid at 1200 UTC 7 Nov 1996. The Qs convergence is contoured and shaded as in Fig. 1b. (c) The 500±900-hPa column-averaged QVR vectors and QVR convergence from a 12-h forecast by the
UW-NMS valid at 1200 UTC 7 Nov 1996. The QVR convergence is contoured and shaded as in Fig. 1b. Surface frontal analysis as in Fig.
2a. (d) The 500±900-hPa column-averaged QDR vectors and QDR convergence from a 12-h forecast by the UW-NMS valid at 1200 UTC 7
Nov 1996. The QDR convergence contoured and shaded as in Fig. 1b. Surface frontal analysis as in Fig. 2a. extreme northeast Iowa to northern Missouri at this time. The 950-hPa at 1200 UTC 7 November 1996 is
The total Qs forcing is shown in Fig. 1b while the QVR shown in Fig. 2a along with the subjectively determined, and QDR contributions are shown in Figs. 1c and 1d, model-based surface frontal analysis at that time. The respectively. The QVR convergence maximum was lo- total Qs forcing (Fig. 2b) is once again composed of a cated north and west of the surface occluded front (Fig. vorticity contribution that is located to the northwest of
1c), while the QDR convergence maximum was nearly the triple point (Fig. 2c) and a deformation contribution coincident with the surface occluded front, ending rather that is collocated with the surface occluded front (Fig. abruptly at the triple point in southwest Wisconsin. Fur- 2d). It is also interesting to note that the region of forcing ther, the QDR convergence maximum is produced by QDR for vertical motion associated with the geostrophic vor- vectors of roughly the same magnitude but opposing sà ticity is broad and weak, whereas the forcing for directions, whereas the QVR convergence maximum associated with the deformation is much longer than it (which is of smaller magnitude but larger areal extent) is wideÐsuggesting a frontal-type scale. Also, the vor- is the result of a diminished magnitude in the vectors, ticity forcing for is located over the sea level pressure not a change in their direction along the sÃ-axis. Impor- minimum. As was the case with the previous example, tantly, this suggests that the QDR vector ®eld forces op- the QDR forcing abruptly ends at the triple point on the posing rotations of equal magnitude to the component southern tip of James Bay. baroclinic zones constituting the sides of the thermal The most vigorous of the three systems described by ridge. Martin (1999) occurred on 1 April 1997. The 950-hPa OCTOBER 1999 MARTIN 2409
along with the model-based subjectively determined nearly circular and located north and west of the peak surface fronts for 0600 UTC 1 April 1997 are shown of the warm sector. Very little convergence of QDR was in Fig. 3a. The dashed line at the western end of the evident at this time (Fig. 4d) although the QDR vectors occluded front indicates a pressure trough that connects themselves were of opposite sà directions along the cold the sea level pressure minimum to a developing sec- frontal and warm frontal zones, respectively. ondary cold front off the Carolina coast. The total Qs In the ensuing 3 h the lower-tropospheric thermal forcing at this time is shown in Fig. 3b. The vorticity structure was altered subtly. An incipient thermal ridge contribution to this total forcing is shown in Fig. 3c. had developed, extending northwestward from the peak Consistent with the prior cases, this forcing is located of the warm sector to Long Island (Fig. 5a). This oc- west of the triple point along the western edge of the cluded thermal ridge, in contrast to the thermal ridge surface occluded front and is the result of a diminishing located over southern Quebec, connected the surface magnitude of the QVR vectors, not a systematic change warm sector to the minimum in sea level pressure. The in their direction along the sà axis. The vorticity forcing surface cyclone center (now at 993 hPa) had moved is also, once again, ellipsoidal in shape with major and northeastward to a position south of Long Island and minor axes of similar lengths. It is also located just to the Qs forcing for ascent was maximized in its vicinity the northwest of the sea level pressure minimum as in (Fig. 5b). The QVR contribution to Qs still provided the the other cases. The QDR vectors themselves are of near- majority of the total Qs forcing and remained to the ly equal magnitude but different sà direction across the north and west of the peak of the warm sector, displaced surface occluded front resulting in a long and narrow slightly west of the position of the sea level pressure but very intense region of QDR convergence located minimum at this time (Fig. 5c). The deformation forcing along the surface occluded front. Also, consistent with and QDR vectors are shown in Fig. 5d. Notice the QDR the other two cases, the QDR forcing for ascent ends vectors change sà direction across the incipient thermal abruptly at the triple point. ridge and their convergence is a maximum over its lim- The remarkable similarity in the distribution of the ited length, ending at the peak of the warm sector. The
QVR and QDR vectors and their respective forcings for change of sà direction of the QDR vectors physically rep- vertical motion in these three quite different cyclones resents the action of the geostrophic deformation in dif- testi®es to the robust nature of the signal and suggests ferentially rotating the component baroclinic zones its relevance for understanding the process of occlusion. (warm and cold fronts) of this cyclone as the storm Before offering a physical interpretation of these in- develops. stantaneous distributions, we proceed to examine the By 0000 UTC 1 April the surface cyclone had inten-
QVR and QDR vectors, along with their convergences, si®ed to 985 hPa and was located south of Cape Cod, throughout the life cycle of the 1 April 1997 cyclone. Massachusetts. The 950-hPa demonstrates that by this time the cyclone had occluded (Fig. 6a) as a thermal ridge extended westward from the peak of the warm 4. Evolution of the separate vorticity and sector into the sea level pressure minimum. The total deformation forcings in an occluding cyclone Qs forcing was notable at this time and was maximized In this section we examine the evolution of the com- from the Delmarva peninsula to well offshore to the ponents of the Qs forcing throughout a portion of the southeast of Cape Cod (Fig. 6b). Partitioning of this life cycle of the 1 April 1997 cyclone. At each time to forcing into its separate vorticity and deformation con- be shown, the Qn component of the forcing for in the tributions provides evidence of an emerging pattern. The developing occluded quadrant is much smaller than the vorticity forcing was distributed over a large area on
Qs component. As is the case in Figs. 1±3, we show the northwestern edge of the new occluded surface front the UW-NMS model-based, subjectively determined (Fig. 6c). The maximum QVR convergence remained surface frontal positions in each of the foregoing ®gures slightly north and west of the sea level pressure mini- along with the 500±900-hPa column-averaged Qs, QVR, mum, suggesting, in accord with theory, that the east- and QDR vectors at the indicated times. ward progression of the surface cyclone was being re- At 1500 UTC 31 March, a modest sea level pressure tarded by the column stretching associated with the as- minimum was located just offshore of New Jersey and cent provided by the vorticity component of the Qs forc- the Delmarva peninsula. The central pressure was 997 ing (i.e., thermal wind advection of geostrophic hPa and the lower-tropospheric frontal structure was vorticity). characteristic of an open wave (Fig. 4a). The Qs vectors The deformation forcing and the QDR vectors are and their convergence were largest just to the northwest shown in Fig. 6d. The QDR vectors change sà direction of the sea level pressure minimum with signi®cant cy- across the surface occluded front and thereby provided -١ along the cold frontal baroclinic a narrow, frontal-scale axis of convergence aligned near clonic rotation of zone suggested by the Qs vectors themselves (Fig. 4b). ly along the surface occluded front, ending at the triple ١ implied by the The vast majority of the Qs convergence and the cy- point. The differential rotation of ١ was accounted for by the vorticity distribution of QDR vectors forced a squeezing of the clonic rotation of contribution (Fig. 4c). Note that the QVR forcing was component frontal zones into a thermal ridge connecting 2410 MONTHLY WEATHER REVIEW VOLUME 127
FIG. 3. (a) As for Fig. 2a except from an 18-h forecast by the UW±NMS valid at 0600 UTC 1 Apr 1997. Dashed cold frontal symbol represents the position of the secondary surface cold front. Bold dashed line represents the position of a pressure trough. (b) As for Fig. 2b except from an 18-h forecast by the UW-NMS valid at 0600 UTC 1 Apr 1997. The Qs convergence is contoured and shaded in units of m kgϪ1 sϪ1 every 10 ϫ 10Ϫ16 mkgϪ1 sϪ1 beginning at 5 ϫ 10Ϫ16 mkgϪ1 sϪ1. (c) As for Fig. 2c except from an 18-h forecast by the UW-NMS valid at 0600 UTC 1 Apr 1997. The QVR convergence is contoured and shaded as in Fig. 3b. Frontal symbols as in Fig. 3a. (d) As for Fig.
2d except from an 18-h forecast by the UW-NMS valid at 0600 UTC 1 Apr 1997. The QDR convergence is contoured and shaded as in Fig. 3b. Frontal symbols as in Fig. 3a. the triple point to the sea level pressure minimum. Thus, had become narrow and intense by this time and closely a lengthening and sharpening of the occluded thermal corresponded to the position of the surface occluded ridge, as well as the vertical motion necessary for the front (Fig. 7b). The vorticity contribution to this forcing production of the clouds and precipitation that charac- was, at this time, located well to the west of the triple terized it, were simultaneous results of the geostrophic point and just north and west of the sea level pressure (nonfrontogenetic) deformation forcing in the vicinity minimum, as had been the case at every stage of the of the thermal ridge. cyclone's evolution.
A similar four-panel diagram corresponding to 0600 The QDR vectors and their convergence at this time UTC 1 April has been shown and described earlier (Fig. are shown in Fig. 7d. The horizontal scale of the QDR 3) and ®ts the exact same pattern as we have described convergence maximum had steadily shrunk during the for the other times presented in this section. By 1200 evolution of this occluded cyclone and was by this time UTC 1 April the surface occluded thermal ridge had very narrow, aligned along the surface occluded front, lengthened considerably and the sea level pressure min- ending at the triple point just as it had done at all times imum (now at 980 hPa) had correspondingly become subsequent to the original development of the thermal further removed from the triple point (Fig. 7a). A sec- ridge. By this stage in the cyclone life cycle, however, ondary cold frontal zone had also become quite obvious the QDR vectors and their convergence constituted the in the lower-tropospheric thermal ®eld. The Qs forcing vast majority of the total Qs forcing. In other words, OCTOBER 1999 MARTIN 2411
FIG. 4. (a) As for Fig. 3a except from a 3-h forecast by the UW-NMS valid at 1500 UTC 31 Mar 1997. (b) As for Fig. 3b except from a 3-h forecast by the UW-NMS valid at 1500 UTC 31 Mar 1997. (c) As for Fig. 3c except from a 3-h forecast by the UW-NMS valid at 1500 UTC 31 Mar 1997. (d) As for Fig. 3d except from a 3-h forecast by the UW-NMS valid at 1500 UTC 31 Mar 1997. late in the life cycle, the nonfrontogenetic, geostrophic level pressure minimum are forced by local upward ver- deformation exerted the largest in¯uence on the total tical motions. In prior work concerning the Q±G forcing Q±G forcing for ascent in the occluded quadrant of the for ascent in the occluded sector of cyclones, Martin cyclone. Once again, the QDR vectors changed sà direc- (1999) showed that the overwhelming majority of the tion across the occluded front, implying the importance lower- and middle-tropospheric forcing for ascent there of the geostrophic deformation in lengthening the ther- is provided by convergence of the along-isentrope com- mal ridge by squeezing the component baroclinic zones ponent of the Q vector, Qs. It was also shown that the together along the QDR convergence maximum. development of the thermal ridge characteristic of oc- cluded cyclones was a direct consequence of conver- gence of Q since Q physically describes the contri- 5. Discussion s s ١. In bution of the geostrophic ¯ow to the rotation of Synoptic experience demonstrates that as a cyclone this study, the separate vorticity and deformation con- occludes, two characteristic transformations occur in the tributions to the Qs vector have been isolated in the lower troposphere: 1) the occluded thermal ridge, which separate vector expressions, QVR and QDR respectively. joins the sea level pressure minimum to the triple point, The foregoing partitioning of Qs illustrates that a char- ®rst develops and then lengthens with time, and 2) the acteristic distribution of these separate component forc- sea level pressure minimum ``retreats'' to the north and ings is involved in the development of an occlusion, west of the triple point, often developing into the cold suggesting that different, but complementary, roles are air. Both the clouds and precipitation that accompany played by the geostrophic vorticity and deformation in the occluded thermal ridge and the behavior of the sea this process. In fact, it appears that each component of 2412 MONTHLY WEATHER REVIEW VOLUME 127
FIG. 5. (a) As for Fig. 3a except from a 6-h forecast by the UW-NMS valid at 1800 UTC 31 Mar 1997. (b) As for Fig. 3b except from a 6-h forecast by the UW-NMS valid at 1800 UTC 31 Mar 1997. (c) As for Fig. 3c except from a 6-h forecast by the UW-NMS valid at 1800 UTC 31 Mar 1997. (d) As for Fig. 3d except from a 6-h forecast by the UW-NMS valid at 1800 UTC 31 Mar 1997.
the forcing plays a central role in one of the two struc- strophic deformation according to the distribution of QDR tural transformations mentioned above. vectors and their convergence. Throughout the evolution
Throughout the evolution depicted in Figs. 4±7, the described in section 4, the QDR vectors changed sign (sà QVR forcing was located to the north and west of the direction) across the occluded thermal ridge. Thus, the peak of the warm sector, just northwest of the sea level cold and warm frontal baroclinic zones, which constitute pressure minimum. The vertical motion that accompa- the two sides of the thermal ridge, were rotated with nies this component of the baroclinic rotation provides approximately equal magnitude but opposite direction by a mechanism for the development of the sea level pres- the geostrophic deformation ®eld. This differential ro- sure minimum and its gradual retreat, during the occlu- tation, the result of convergence of the QDR vectors near sion process, into the cold air north and west of the the thermal ridge, squeezed the warm and cold frontal original warm sector. Thus, as ®rst suggested by Sut- baroclinic zones together, thereby sharpening and length- cliffe (1947) and later reiterated by Trenberth (1978), ening the thermal ridge and promoting the development the evolution of the sea-level pressure minimum is con- of the occluded front. The QDR convergence provided trolled throughout the cyclone life cycle by the thermal forcing for ascent of the warm sector air that was wind advection of geostrophic vorticity. squeezed between the two baroclinic zones during the The sharpening and lengthening of the occluded ther- process of occlusion. This process is illustrated in sche- mal ridge and the associated development of the occluded matic form in Fig. 8. It was predominantly this forcing front observed in Figs. 4±7, along with the persistence for ascent that accounted for the clouds and precipitation of cloudiness and precipitation in its vicinity (not shown) that characterized the occluded quadrant of this cyclone are simultaneously forced by the nonfrontogenetic geo- (Martin 1999). OCTOBER 1999 MARTIN 2413
FIG. 6. (a) As for Fig. 3a except from a 12-h forecast by the UW-NMS valid at 0000 UTC 1 Apr 1997. (b) As for Fig. 3b except from a 12-h forecast by the UW-NMS valid at 0000 UTC 1 Apr 1997. (c) As for Fig. 3c except from a 12-h forecast by the UW-NMS valid at 0000 UTC 1 Apr 1997. (d) As for Fig. 3d except from a 12-h forecast by the UW-NMS valid at 0000 UTC 1 Apr 1997.
Another relevant feature of the evolution of the sep- ing was twice as large as the maximum QVR forcing. arate QVR and QDR forcings is illustrated in Figs. 4±7. The fact that the increasing magnitude of the QDR forc- Early in the life cycle (Fig. 4) the developing cyclone ing occurred simultaneously with the development of was characterized by a broad warm sector and little the thermal ridge is consistent with the recent study of evidence of a thermal ridge at the peak of the warm the deformation term in the Q±G omega equation by sector. At this stage, QVR convergence described nearly Martin (1998a). He showed that the deformation term all of the total Qs convergence, suggesting that in the can be large, even in the middle troposphere, in regions early development stage geostrophic deformation plays where second derivatives of are superposed with non- a minimal role in baroclinic rotation and its associated zero ®rst derivatives of the geostrophic wind ®eld. The vertical motion forcing near the cyclone center. Even at thermal ridge in the occluded quadrant of the 1 April 1800 UTC 31 March (Fig. 5), by which time a more cyclone possessed both of these characteristics. pointed peak to the warm sector had developed, the Qs Based upon the detailed analysis of the 1 April case forcing was still dominated by the QVR forcing, although presented here, and the analysis of several other cases a modest contribution by the QDR forcing was precisely examined in the course of this research, we propose the collocated with the nascent thermal ridge at that time. following dynamical interpretation of the occlusion pro- The subsequent evolution of the separate components cess. During the early development stage of a midlati- of Qs demonstrated the growing importance of QDR con- tude cyclone, an upper-level vorticity anomaly en- vergence to the total Q±G vertical motion forcing in the croaches upon a lower-tropospheric baroclinic zone occluded quadrant (Figs. 6, 3, and 7). By 1200 UTC 1 (Fig. 9a). Petterssen and Smebye (1971) described this
April (Fig. 7) the magnitude of the maximum QDR forc- as a ``Type B'' development and it is controlled by 2414 MONTHLY WEATHER REVIEW VOLUME 127
FIG. 7. (a) As for Fig. 3a except from a 24-h forecast by the UW-NMS valid at 1200 UTC 1 Apr 1997. Dashed cold frontal symbol represents the position of the secondary surface cold front. Bold dashed line represents the position of a pressure trough. (b) As for Fig. 3b except from a 24-h forecast by the UW-NMS valid at 1200 UTC 1 Apr 1997. (c) As for Fig. 3c except from a 24-h forecast by the UW- NMS valid at 1200 UTC 1 Apr 1997. Frontal symbols as in Fig. 7a. (d) As for Fig. 3d except from a 24-h forecast by the UW-NMS valid at 1200 UTC 1 Apr 1997. Frontal symbols as in Fig. 7a.
(QDR ´ thermal wind advection of geostrophic vorticity as geostrophic deformation forcing for ascent (Ϫ2١ shown by Sutcliffe (1947) and Trenberth (1978). The assumes increasing importance as the second derivative region of cyclonic vorticity advection by the thermal of grows in step with the increasing sharpness of the wind is also a region of convergence of QVR with the thermal ridge. Sustained ascent associated with the com- largest QVR vectors occurring in the maximum vorticity bination of vorticity and deformation forcing ensures area (Fig. 9b). These QVR vectors represent the contri- that ®rst derivatives of the geostrophic wind ®eld remain -١. As collocated with the thermal ridge. The differential ro bution of the geostrophic vorticity to rotation of
-١ described by convergence of the QDR vec such, they differentially rotate the lower-tropospheric tation of baroclinic zone and create a nascent thermal ridge in tors results in a squeezing together of the cold and warm the region of ascent and height falls downshear of the frontal baroclinic zones. This process occurs on a small- upper-level vorticity feature (Fig. 9c). Conversely, a er scale than the forcing for ascent provided by con- thermal trough in the region of descent and height rises vergence of QVR and proceeds rapidly to intensify (and is forced upshear of the upper-level vorticity feature. lengthen) the occluded thermal ridge, thereby devel- As soon as the incipient thermal ridge is suf®ciently oping the surface occluded front, through a positive well developed, the deformation forcing (QDR) grows feedback that relates the magnitude of the QDR forcing in magnitude and begins to sharpen and lengthen the to the magnitude of the second derivative in (Fig. 9e). occluded thermal ridge by differentially rotating the As the cyclone continues to evolve past the occluded component baroclinic zones that border it (Fig. 9d). At stage, the QVR forcing weakens as vorticity advection this stage in the cyclone life cycle, the nonfrontogenetic lessens. This results in weaker height falls and a weak- OCTOBER 1999 MARTIN 2415
tionless, adiabatic ¯ow the dynamics of this PV wave breaking could be investigated in a manner similar to
that used here by ®rst calculating a vector ®eld, QPV, de®ned as the Lagrangian rate of change of the PV gradient vector following the geostrophic motion:
Vץ Vץ d . ˆPV)j)١ ´ PV)i,Ã Ϫ)١ ´١PV ϭϪ gg Q ϭ yץ xץ PV dt g [] (11) The development and use of such a diagnostic has re- cently been discussed by Davies and Rossa (1998). A
partition of QPV into its components,QQPVVR and PV DR (exactly analogous to the Q-vector partitions, QVR and QDR, developed in this paper), would isolate the effects of geostrophic vorticity and deformation, respectively, FIG. 8. Schematic illustrating the role of geostrophic deformation in sharpening and lengthening the occluded thermal ridge. Dark on the morphological changes in tropopause PV, which (light) solid lines represent column-averaged isentropes at the initial are central to the processes of cyclogenesis and occlu-
(later) time. The QDR vectors are valid at the initial time. The potential sion. Such an investigation is currently under way and temperature gradient vectors (dark arrows for initial time, light arrows should provide a new interpretation of, and new insights for later time) on either side of the thermal ridge axis are rotated in into, aspects of the dynamics and kinematics of the mid- the direction of Q . DR latitude cyclone life cycle. ened geostrophic deformation ®eld in the vicinity of the 6. Conclusions thermal ridge. This, in turn, leads to weaker QDR forcing and an eventual halt to forcing for ascent in the aging In this paper separate vector expressions representing occluded sector. the contributions of the geostrophic vorticity and de-
It was suggested by K92 that a scale separation exists formation to the along-isentrope component (Qs) of the between the forcing for ascent associated with the along- Q vector have been isolated. It has been shown that the
-١ made by the geo and across-isentrope components of Q, Qs, and Qn, re- contribution to the rotation of spectively. The analyses presented here suggest that strophic vorticity can be represented by a vector, QVR, some of the frontal-scale Q±G forcing for ascent, name- the convergence of which is precisely equal to the ther- ly, the QDR forcing, appears to reside within the along- mal wind advection of geostrophic vorticity, a funda- isentrope component of Q. It seems more appropriate, mental dynamical quantity in midlatitude synoptic de- therefore, to suggest instead that an identi®able scale velopment (Sutcliffe 1947; Trenberth 1978). A parti- separation exists between the forcings for ascent asso- tioning of Qs into its vorticity (QVR) and deformation ciated with geostrophic vorticity (QVR) and geostrophic (QDR) contributions in three different cyclones, along deformation (Qn and QDR). Recognition of this scale with an analysis of their respective evolutions in a single separation is implicit in the pioneering works of Pet- occluded cyclone, were undertaken. The results of this terssen (1936) and Sutcliffe (1947), but is made partic- analysis suggest a dynamical explanation of two com- ularly clear in terms of the foregoing Q-vector parti- monly observed structural transformations associated tioning. with occluded midlatitude cyclones. Finally, the deformation of contours observed dur- First, the tendency for the sea level pressure minimum ing occlusion suggests that the occlusion process can to deepen and migrate northward and westward, into the be viewed as a tropospheric thermal wave-breaking cold air, after occlusion is controlled by the convergence event (Hakim et al. 1996), particularly in light of the of QVR. This result reaf®rms the long-standing synoptic de®nition of wave-breaking given by McIntyre and forecasting rule that the behavior of the surface cyclone Palmer (1983)Ð``the rapid and irreversible deformation center is largely controlled by the thermal wind advec- of a material contour.'' In this paper the contributions tion of geostrophic (absolute) vorticity (Sutcliffe 1947). of the geostrophic vorticity and deformation to this ther- Second, convergence of QDR along the thermal ridge mal wave-breaking have been isolated through use of axis differentially rotates the cold and warm frontal bar- the Q vector, which relates geostrophic wind ®elds to oclinic zones that border that axis, rapidly sharpening the conservative (material) variable, . The occlusion and lengthening the occluded thermal ridge and leading process also involves a characteristic tropopause poten- to the development of the surface occluded front. This tial vorticity (PV) structure, referred to as the ``treble convergence forces a narrow, frontal-scale region of as- clef'' by Martin (1998b). The development of this tro- cent that lifts the warm sector air that is squeezed be- popause PV structure results from the breaking of a tween the cold and warm frontal zones during this pro- tropopause PV wave. Since PV is conserved in fric- cess. This lifting is made manifest in the clouds and 2416 MONTHLY WEATHER REVIEW VOLUME 127
FIG. 9. Schematic illustration of the separate roles played by geostrophic vorticity and deformation in the process of occlusion. (a) Effect of an upper-level vorticity (PV) maximum on development at an initial time, t ϭ 0. Thick solid lines are column average isentropes, thin solid lines are midtropospheric vorticity (PV) contours with ``ϫ'' signifying a cyclonic vorticity maximum. Shading indicates an area of cyclonic vorticity advection (CVA) by the thermal wind. (b) As in Fig. 9a except thick solid arrows are QVR vectors and the shading represents the QVR convergence maximum at t ϭ 0. (c) Dashed lines are column average isentropes at time t ϭ t1 after rotation by the QVR vectors in
Fig. 9b. Incipient thermal ridge is located in the region of QVR convergence maximum. (d) Close-up of the thermal ridge in Fig. 9c at t ϭ t1. The QDR vectors converge on the thermal ridge axis and force a QDR convergence maximum (shaded) of frontal scale. (e) Thermal ridge at time t ϭ t2 intensi®ed through differential rotation implied by QDR vectors in Fig. 9d. The QDR vectors at t ϭ t2 are larger in response to a larger second derivative of in the vicinity of the thermal ridge. Shading represents QDR convergence maximum at t ϭ t2. precipitation that extend along the occluded thermal squeezing together of the warm and cold frontal zones ridge from the triple point to the sea level pressure min- and a resultant lifting of warm sector air. imum. Thus, the distribution of QDR vectors and their The geostrophic vorticity and deformation contribu- convergence in the occluded sector provide both a re- tions to baroclinic rotation provide signi®cant, spatially af®rmation and dynamical explanation of the traditional separate forcings for ascent in the occluded quadrant, synoptic view that the occlusion process involves the which, taken together, account for nearly all of the Q±G OCTOBER 1999 MARTIN 2417 forcing for ascent in the occluded sector of cyclones. ץ ץ Vץ ץUggץ The dynamical model of the occlusion process that QDR ϭ f o␥ ϪϪ Ϫ yץ yץ xץ xץ xץemerges from this work involves two steps. First, the Ά [ geostrophic vorticity forces cyclonic rotation of the cold ץ ץ Vץ ץ Uץ frontal baroclinic zone, production of a nascent thermal ϩϪgg Ϫ [ xץ yץ yץ xץ yץridge, and ascent that initially deepens the surface cy- clone. Once the thermal ridge exists, the nonfrontoge- 22 (١ (kˆ ϫץ ץ UץVggץ 1 netic geostrophic deformation, whose magnitude is de- Ϫ f o␥ Ϫϩ . ١|2| ·[ yץ xץ ]y ץ xץpendent on the magnitude of the second derivative of 2 , attains greater signi®cance. The increased deforma- tion forcing squeezes the cold and warm frontal baro- clinic zones together, sharpening and lengthening the Carrying out all the multiplications and simplifying the thermal ridge, leading to the development of the char- resulting expression leads to acteristic occluded thermal structure. This process is 2ץ Vץ 1ץ ץ Vץ Uץ accompanied by forcing for ascent (convergence of Q ϭ f ␥ ggϪϩ g yץ xץ y 2ץ xץ yץ xץ DR o QDR) that occurs on a frontal scale but is a dynamical [ consequence of the baroclinic rotation forced by the ץ Vץ 221ץ Uץ 1 geostrophic deformation. ϪϪgg xץx ץ x 2ץy ץ The results presented here introduce a re®nement to 2 an emerging dynamical interpretation of the occlusion 2 (١ (kˆ ϫץ Ugץ process, a process traditionally considered to signal the 1 ϩ . (A3) ١|2| [ yץy ץ commencement of cyclone decay. We are currently em- 2 ploying a similarly Q-vector partition to examine the thermal devolution of decaying cyclones within the con- Substituting the expressions for the stretching and shear- text of a broader dynamical investigation of cyclolysis. ing deformations, E1 and E 2, respectively, where E1 ϭ y) yieldsץ/Ugץx ϩץ/Vgץ) y) and E 2 ϭץ/Vgץx Ϫץ/Ugץ)
ץ 221ץ 1ץ ץ Acknowledgments. The author wishes to thank Drs. Q ϭ f ␥ E ϩ E Ϫ E xץy 2 2 ץy 2 ץ xץDaniel Keyser, Juan Carlos Jusem, Frederick Sanders, DR o []12 and two anonymous reviewers for numerous thought- (١ kˆ ϫ) provoking comments that helped to improve the man- ϫ ١|2| uscript. This work was funded by the National Science Foundation under Grant ATM-9505849. It is dedicated or, finally, to the memory of Mr. Du'o'ng Nhu LaÃm, father-in-law ץ 22ץ 1ץ ץ ␥ of the author. f .١ Q ϭ o E ϩ E Ϫ kˆ ϫ · xץy ץy 2 2 []ץ xץ١|2 Ά 1|DR (A4) APPENDIX A APPENDIX B Derivation of the Cartesian Expression for QDR An Alternative Derivation of QVR and Q⌬R 1 Recall that QDR ϭ Qs Ϫ 2QTR. Therefore, A similar partition of Qs into its deformation and vorticity components can be obtained following the de- scription given in Davies-Jones (1991). Beginning with ١)1 ١)(kˆ ϫ Q ´(kˆ ϫ ١) 11 QDR ϭϪ( fog␥ )(kˆ ϫ (١|2 ] Q ϭ (Q ϩ D) ϩ (Q Ϫ D), (B1||١|[]] 22 or, equivalently, [(١´(yץ/ Vץ)١), (Ϫ´(xץ/ Vץ)where Q ϭ f ␥[(Ϫ ١) o g g kˆ ϫ)1 -١)]. The con ´ ١), (Ϫ١Vg ´ ١|,2 and D ϭ f o␥[(Ϫ١Ug|( ١) Ϫ ( f ␥ Q ϭ Q ´(kˆ ϫ ١|2 vergence of D physically represents the Q±G forcing DR []2|og (A1) for ascent from the deformation term neglected in the approximate Trenberth (1978) forcing for omega (i.e., . Q ´ Q ϩ 2١ ´ D ϭϪ2١ ´ D ϭ Q Ϫ Q so that Ϫ2١ Ã TR TR ١)ˆj] Upon expansion of (B1), the components of the Q´(yץ/Vgץ)١)i, (Ϫ´(xץ/Vgץ)where Q ϭ f o␥[(Ϫ x)ˆj). Thus, vector becomeץ/ץ) ,y)Ãiץ/ץ)١ ϭ (Ϫ and kˆ ϫ 2418 MONTHLY WEATHER REVIEW VOLUME 127
ץ 22ץ 1ץ ץ ␥ fץ Uץ Vץ 1ץ Uץ Q ϭ f ␥ ϪϪϩggg Q ϭ o E ϩ E Ϫ · xץy ץy 2 ץ xץ١|2 ΆΆ[]12| y sץ yץ xץx 2ץ xץ ]xo ␥ fץ Uץ Vץ 1 (١), (B5 ١) ϩ o (kˆ ϫ ϪϪgg (B2a) ϫ(kˆ ϫ y] · 2 gץ yץ xץ2 and which is identical to the form described in this paper with the ®rst term in (B5) equal to QDR and the second term equal to QVR. The form of Q in (B4) can also beץ Ugץ Vץ 1ץ Vץ Q ϭ f ␥ ϪϪgg ϩ x used to show that the across-isentrope component of Qץ yץ xץy 2ץ yץ ]yo contains no contribution from geostrophic vorticity as ١| ) is given by|/١)(|١|/١ ´ Qn ϭ (Qץ Uץ Vץ 1 ϩϪgg. (B2b) x 22ץ yץ xץ 2 ץ ץ ץ ץ ١)1)␥] f o Q ϭ E ϪϪE . (B6) n 2 12 yץ xץ · xץy ץ١|2[]Ά| xץ/ Vץ) y) and E ϭץ/ Uץx Ϫץ/ Vץ) Now, we let ϭ g g g 2 g y) to represent the geostrophic vorticity andץ/Ugץϩ shearing deformation, respectively. With these substi- REFERENCES tutions, the component expressions for Q reduce to Barnes, S. L., and B. R. Colman, 1993: Quasigeostrophic diagnosis of cyclogenesis associated with a cutoff extratropical cycloneÐ . The Christmas 1987 storm. Mon. Wea. Rev., 121, 1613±1634ץ 1ץ 1ץ Uץ Q ϭ f ␥ ϪϪg E Ϫ (B3a) Davies, H. C., and A. M. Rossa, 1998: PV frontogenesis and upper .y tropospheric fronts. Mon. Wea. Rev., 126, 1528±1539ץ y 2 gץ x 2 2ץ xץxo Davies-Jones, R., 1991: The frontogenetical forcing of secondary and circulations. Part I: The duality and generalization of the Q vec- tor. J. Atmos. Sci., 48, 497±509. Hakim, G., L. Bosart, and D. Keyser, 1996: The Ohio Valley wave- ץ 1ץ 1ץ Vgץ Qyoϭ f ␥ ϪϪE2 ϩ g . (B3b) merger cyclogenesis event of 25±26 January 1978. Part II: Di- .x agnosis using quasigeostrophic potential vorticity inversionץ x 2ץ y 2ץ yץ Mon. Wea. Rev., 124, 2176±2205. The stretching derivative in the ®rst term of each Holton, J. R., 1992: An Introduction to Dynamical Meteorology. 3d expression in (B3) can be decomposed in a similar way ed. Academic Press, 511 pp. Hoskins, B. J., and M. A. Pedder, 1980: The diagnosis of middle into latitude synoptic development. Quart. J. Roy. Meteor. Soc., 106, 707±719. V 11ץ U 11ץ ggϭ ␦ ϩ E and ϭ ␦ ϩ E , , I. Draghici, and H. C. Davies, 1978: A new look at the - .y 22 equation. Quart. J. Roy. Meteor. Soc., 104, 31±38ץx 2211ץ Keyser, D., M. J. Reeder, and R. J. Reed, 1988: A generalization of Petterssen's frontogenesis function and its relation to the forcing where of vertical motion. Mon. Wea. Rev., 116, 762±780. , B. D. Schmidt, and D. G. Duffy, 1992: Quasigeostrophic ver- tical motions diagnosed from along- and cross-isentrope com- VץU ggץ VץUggץ ␦ ϭϩ and E1 ϭϪ ponents of the Q vector. Mon. Wea. Rev., 120, 731±741. y Kurz, M., 1997: The role of frontogenetic and frontolytic wind ®eldץ xץy ץ xץ effects during cyclone development. Meteor. Appl., 4, 353±363. represent the geostrophic divergence and stretching de- Martin, J. E., 1998a: On the deformation term in the quasigeostrophic formation, respectively. omega equation. Mon. Wea. Rev., 126, 2000±2007. , 1998b: The structure and evolution of a continental winter Thus, the Q vector can be written in its component cyclone. Part I: Frontal structure and the classical occlusion pro- form as the sum of the geostrophic stretching and shear- cess. Mon. Wea. Rev., 126, 303±328. ing deformations as well as the geostrophic vorticity as , 1999: Quasigeostrophic forcing of ascent in the occluded sector of cyclones and the trowal airstream. Mon. Wea. Rev., 127, 70± 88. McIntyre, M. E., and T. N. Palmer, 1983: Breaking planetary wavesץ 1ץ 1ץ 1 Q ϭ f ␥ Ϫ E Ϫ E Ϫ (B4a) .y in the stratosphere. Nature, 305, 593±600ץ y 2 gץ x 2ץxo2 12 Petterssen, S., 1936: A contribution to the theory of frontogenesis. and Geofys. Publ., 11, 1±27. , and S. J. Smebye, 1971: On the development of extratropical . storms. Quart. J. Roy. Meteor. Soc., 97, 457±482ץ 1ץ 1ץ 1 Q ϭ f ␥ E Ϫ E ϩ . (B4b) Sutcliffe, R. C., 1947: A contribution to the problem of development. .x Quart. J. Roy. Meteor. Soc., 73, 370±383ץ x 2 gץ y 2ץyo2 12 Trenberth, K. E., 1978: On the interpretation of the diagnostic quasi- Given the form of Q in (B4), it is easily shown that geostrophic omega equation. Mon. Wea. Rev., 106, 131±137. Wiin-Nielsen, A., 1959: On a graphical method for an approximate .١|)]is determination of the vertical velocity in the mid-troposphere|/١ ١|)(kˆ ϫ|/١ Qs [where Qs ϭ (Q ´ kˆ ϫ given by Tellus, 11, 432±440.