The Separate Roles of Geostrophic Vorticity and Deformation in the Midlatitude Occlusion Process
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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 ducing secondary circulations to which the character- differential rotation of =u implied by the convergent 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 of =u and called them Fn and Fs, respectively. K92 s demonstrated that the corresponding quasigeostrophic vector expressions representing the rates of change of direction of =u produced by the geostrophic vorticity 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 q 1999 American Meteorological Society OCTOBER 1999 MARTIN 2405 and precipitation distribution, of the postmature phase ]]2 V midlatitude cyclone. s¹122f v 2 f g ´ =z (2a) oog2 ø The analysis begins with a brief review of the various 12]p ]p 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 ]2 The traditional form of the Q±G omega equation is 22 s¹1f o v 522= ´ Q, (3) given by (Holton 1992) 12]p2 where Q is given by ]2 s¹122f v o ]p2 ]V ]V 12 Q 5 f g 2 gg´ =u i,à 2 ´ =u jˆ o []1212]x ]y ]]f 2 5 f [V ´ =(z 1 f )] 2¹ V ´ = , (1) c /c og g g with g 5 (R/fP )(P /P) y p . Thus, the divergence of the ]p []12]p oo Q vector describes the complete forcing for the Q±G 2 2 2 2 2 2 where ¹ 5] /]x 1] /]y , zg 5 (1/ f o)¹ f, and ]f/]p omega equation. Another important physical meaning 52RT/p. It is tempting to consider the two terms on of the Q vector is that it represents the rate of change the rhs of (1) as representations of separate physical of =u following the geostrophic ¯ow [i.e., Q 5 processes; however, a convincing argument against such f og(d/dtg)=u, where d/dtg 5]/]t 1 Ug ]/]x 1 Vg ]/]y]. a practice was made by Trenberth (1978) and Hoskins As such, Q describes changes in the magnitude of =u et al. (1978). Upon carrying out the derivatives on the as well as changes in the direction of =u. For this reason, 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 ]2 nà is directed along =u and sà is 908 counterclockwise s¹122f v o 2 from nà (slightly different from that used by K88 and 12]p K92). In such a coordinate system the across- and along- isentrope components of Q (Q and Q , respectively) ]Vg n s 5 2 f og´ =(z 1 f ) are given by ]p A Q ´ =u =u Q 55Q nà (4a) nn|=u||=u| ]Vgggg]U ]V ]V [] 1 2 f ´ = 2 ´ = , (2) o []]y 12]p ]x 12]p and B Q ´(kˆ 3 =u)(kˆ 3 =u) Q 55Q sÃ. (4b) which is approximately equal to ss|=u||[]=u| 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.