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Using Tropopause Maps to Diagnose Midlatitude Weather Systems OCTOBER 1998 MORGAN AND NIELSEN-GAMMON 2555 Using Tropopause Maps to Diagnose Midlatitude Weather Systems MICHAEL C. MORGAN Department of Atmospheric and Oceanic Sciences, University of WisconsinÐMadison, Madison, Wisconsin JOHN W. N IELSEN-GAMMON Department of Meteorology, Texas A&M University, College Station, Texas (Manuscript received 23 June 1997, in ®nal form 28 November 1997) ABSTRACT The use of potential vorticity (PV) allows the ef®cient description of the dynamics of nearly balanced at- mospheric ¯ow phenomena, but the distribution of PV must be simply represented for ease in interpretation. Representations of PV on isentropic or isobaric surfaces can be cumbersome, as analyses of several surfaces spanning the troposphere must be constructed to fully apprehend the complete PV distribution. Following a brief review of the relationship between PV and nearly balanced ¯ows, it is demonstrated that the tropospheric PV has a simple distribution, and as a consequence, an analysis of potential temperature along the dynamic tropopause (here de®ned as a surface of constant PV) allows for a simple representation of the upper-tropospheric and lower-stratospheric PV. The construction and interpretation of these tropopause maps, which may be termed ``isertelic'' analyses of potential temperature, are described. In addition, techniques to construct dynamical representations of the lower-tropospheric PV and near-surface potential temperature, which complement these isertelic analyses, are also suggested. Case studies are presented to illustrate the utility of these techniques in diagnosing phenomena such as cyclogenesis, tropopause folds, the formation of an upper trough, and the effects of latent heat release on the upper and lower troposphere. 1. Introduction (Shapiro and Grell 1994), but the evolution of VIP is not governed by a single two-dimensional wind ®eld. For potential vorticity (PV) to be qualitatively applied Bleck (1990) proposed the three-dimensional depic- in describing the evolution of tropospheric synoptic- tion of an isosurface of isentropic vorticity (the vertical scale ¯ows, the three-dimensional PV and surface po- component of absolute vorticity computed from winds tential temperature distributions must be simply repre- along an isentropic surface). Bleck's technique has the sented and easily interpreted. Toward this goal, Hoskins advantage of combining upper- and lower-tropospheric et al. (1985, HMR hereafter) have advocated the con- information onto a single map, but with a resulting loss struction of isentropic PV analyses and surface potential of precision as to the locations and amplitudes of various temperature maps to describe and represent dynamical features. Also, unlike PV, isentropic vorticity is not con- processes within the atmosphere. By plotting Ertel PV served following adiabatic, frictionless ¯ow, so that the on an isentropic surface, one has the advantage of view- evolution of the isentropic vorticity ®eld is dif®cult to ing the dynamics of a conserved variable along the sur- infer from an instantaneous depiction. Alternatively, face of another conserved variable. Thus, with the wind three-dimensional isosurfaces of Ertel PV have been ®eld superposed on an isentropic analysis of PV, one utilized by several authors (including Uccellini 1990; may deduce the conservative (advective) time tendency Lamarque and Hess 1994; Rotunno et al. 1994) to depict of the PV. A distinct disadvantage of this technique is the structure of the tropopause during tropopause fold that several isentropic surfaces must be viewed in order events. This provides a clear depiction of the distribu- to apprehend the full three-dimensional distribution of tion of PV, but like Bleck's method, it is dif®cult to PV and PV advection. Alternatively, the mean PV with- incorporate wind information to infer the evolution. in all layers intersecting the tropopause, or VIP (ver- The focus of this paper is the analysis of both po- tically integrated potential vorticity), may be plotted tential temperature and wind along a two-dimensional projection of the dynamic tropopause (de®ned as a sur- face of constant Ertel PV). This analysis (here referred to as a tropopause map) exploits the rather simple dis- Corresponding author address: Prof. Michael C. Morgan, De- partment of Atmospheric Sciences, University of WisconsinÐMad- tribution of tropospheric PV: as will be demonstrated ison, 1225 West Dayton St., Madison, WI 53706. in the next section, isentropic gradients of PV are con- q 1998 American Meteorological Society 2556 MONTHLY WEATHER REVIEW VOLUME 126 centrated at the tropopause. Tropopause maps are a com- tial, w), a similarly truncated form of the vorticity equa- pact way to represent this distribution. Thus, the essen- tion, a thermodynamic equation, and a mass conser- tial character of the upper-tropospheric PV may be de- vation equation. The initial conditions in a time inte- picted on a single chart, rather than several isentropic gration of a balance system need only be speci®ed by maps. a single variable (Gent and McWilliams 1983). A fa- Tropopause maps, often in conjunction with analyses miliar example is a ¯ow satisfying the quasigeostrophic of lower-tropospheric potential temperature, have been system. As long as z/f 0 .21 (where z is the relative used by many authors in studies of surface cyclogenesis vorticity and f 0 is a characteristic midlatitude value of (e.g., Hoskins and Berrisford 1988; Davis and Emanuel the Coriolis parameter), all ®elds (wind, temperature, 1991; Hakim et al. 1995; Bosart et al. 1996; Bresky and vertical motion, and height tendency) may be expressed Colucci 1996). Thorncroft et al. (1993) used the anal- in terms of the instantaneous geopotential height ®eld. yses to compactly represent the evolution of upper-tro- All ®elds could also, in principle, be speci®ed in terms pospheric PV in a modeling study of life cycles of bar- of the instantaneous temperature ®eld, or even a single oclinic waves. Upper-level systems examined using tro- component of the wind ®eld. popause maps include a tropopause fold (Nielsen et al. The PV can also be used for this purpose. In their 1991), a blocking anticyclone (Morgan et al. 1991), a review, HMR discussed the principles of conservation cutoff cyclone (Bell and Bosart 1993), and mobile and invertibility of PV. The invertibility principle states troughs (Nielsen-Gammon 1995; Lackmann et al. 1997). that the three-dimensional distribution of PV within Wu and Emanuel (1993) and Bosart and Lackmann some domain and boundary conditions on that domain (1995) used the analyses to represent the out¯ow from uniquely determine the velocity and temperature distri- tropical cyclones and the interaction of tropical cyclones bution consistent with a particular balance constraint. with midlatitude systems. The conservation principle states that for inviscid ¯ows Despite the growing use of such analyses, no com- in which the gradient of diabatic heating does not project prehensive discussion of the motivation for these anal- onto the local absolute vorticity vector, PV is conserved yses, nor a guide to constructing or interpreting them, following the ¯uid motion. Together, conservation and has been presented. The intent of this paper is to present invertibility imply that the dynamics of quasi-balanced various techniques for representing the dynamically im- ¯ows may be expressed simply and entirely in terms of portant characteristics of the tropospheric PV distribu- the distribution of PV and the boundary conditions nec- tion in a compact form and to offer an interpretation for essary in the inversion1 of the PV. The use of conser- these representations. The paper is divided into six sec- vation and invertibility to infer the dynamics of quasi- tions, any of which may be skipped without serious loss balanced ¯ows is called ``PV thinking.'' of continuity. In section 2, the theoretical and obser- A simple example of conservation and invertibility vational motivation for tropopause maps is presented. is nondivergent barotropic ¯ow. For such a ¯ow, the Section 3 describes and contrasts the various techniques absolute vorticity and lateral boundary conditions con- one might use to represent the PV near the tropopause tain all of the dynamical information concerning the and the potential temperature and PV distribution in the ¯uid motion. The streamfunction ®eld, which can be lower troposphere. The next two sections are brief case determined from the relative vorticity ®eld through the studies of cyclogenesis that illustrate how tropopause inversion of a two-dimensional elliptic operator, com- maps may be used to diagnose and interpret synoptic- pletely speci®es the velocity ®eld. The balance condi- scale (i.e., quasi-balanced) weather systems. A summary tion that this ¯ow satis®es is nonlinear balance. and further suggestions for how these analyses might In the quasigeostrophic (QG) system, the quasigeo- be utilized is presented in section 6. strophic potential vorticity (hereafter QGPV,also known as the pseudo-PV) is conserved following the geo- strophic ¯ow (Charney and Stern 1962). Quasigeos- 2. Observational and dynamical motivation trophic PV, qp, is de®ned as a. Balanced ¯ow, PV, and PV inversion 1 ] 1 ] q 2 f [¹2w91 f w9 , (2.1) A characteristic of low Rossby number ¯ow in rotating p 0 f0 ]pS12]p ¯uids is the tendency of the ¯ow toward balance. A general type of balance, which subsumes geostrophic balance and where f is the Coriolis parameter, S is a reference state includes
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