QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY VOl. 111 OCTOBER 1985 No. 470 Quart. J. R. Met. Soc. (1985), 111, pp. 877-946 551.509.3:551.511.2:551.511.32 On the use and significance of isentropic potential vorticity maps By B. J. HOSKINS’, M. E. McINTYRE’ and A. W. ROBERTSON3 I Department of Meteorology, University of Reading Department of Applied Mathematics and Theoretical Physics, University of Cambridge Laboratoire de Physique et Chimie Marines, Universitk Pierre et Marie Curie, 75230 Paris Ctdex 05 (Received 12 February 1985, revised 2 July 1985) SUMMARY The two main principles underlying the use of isentropic maps of potential vorticity to represent dynamical processes in the atmosphere are reviewed, including the extension of those principles to take the lower boundary condition into account. The first is the familiar Lagrangian conservation principle, for potential vorticity (PV) and potential temperature, which holds approximately when advective processes dominate frictional and diabatic ones. The second is the principle of ‘invertibility’ of the PV distribution, which holds whether or not diabatic and frictional processes are important. The invertibility principle states that if the total mass under each isentropic surface is specified, then a knowledge of the global distribution of PV on each isentropic surface and of potential temperature at the lower boundary (which within certain limitations can be considered to be part of the PV distribution) is sufficient to deduce, diagnostically, all the other dynamical fields, such as winds, temperatures, geopotential heights, static stabilities, and vertical velocities, under a suitable balance condition. The statement that vertical velocities can be deduced is related to the well-known omega equation principle, and depends on having sufficient information about diabatic and frictional processes. Quasi-geostrophic, semi- geostrophic, and ‘nonlinear normal mode initialization’ realizations of the balance condition are discussed. An important constraint on the mass-weighted integral of PV over a material volume and on its possible diabatic and frictional change is noted. Some basic examples are given, both from operational weather analyses and from idealized theoretical models, to illustrate the insights that can be gained from this approach and to indicate its relation to classical synoptic and air-mass concepts. Included are discussions of (a) the structure, origin and persistence of cutoff cyclones and blocking anticyclones, (b) the physical mechanisms of Rossby wave propagation, baroclinic instability, and barotropic instability, and (c) the spatially and temporally nonuniform way in which such waves and instabilities may become strongly nonlinear, as in an occluding cyclone or in the formation of an upper air shear line. Connections with principles derived from synoptic experience are indicated, such as the ‘PVA rule’ concerning positive vorticity advection on upper air charts, and the role of disturbances of upper air origin, in combination with low-level warm advection, in triggering latent heat release to produce explosive cyclonic development. In all cases it is found that time sequences of isentropic potential vorticity and surface potential temperature charts-which succinctly summarize the combined effects of vorticity advection, thermal advection, and vertical motion without requiring explicit knowledge of the vertical motion field-lead to a very clear and complete picture of the dynamics. This picture is remarkably simple in many cases of real meteorological interest. It involves, in principle, no sacrifices in quantitative accuracy beyond what is inherent in the concept of balance, as used for instance in the initialization of numerical weather forecasts. CONTENTS 1. INTRODUCTION AND HISTORICAL REVIEW l(a) Early ideas l(6) Rossby and Ertel l(c) Subsequent developments l(d) The invertibility principle for potential vorticity 2(a) Preliminaries 2. ISENTROPIC POTENTIAL VORTICITY MAPS FROM ROUTINE ANALYSES 877878 B. J. HOSKINS, M. E. McINTYRE and A. W. ROBERTSON 3. 4. 5. 6. 7. 8. 9. 2(b) Vertical structure and time-variability 2(c) 2(d) A minor blocking episode 2(e) SOME SIMPLE EXAMPLES, FOLLOWING KLEINSCHMIDT Development of a North Atlantic cutoff cyclone The conceptual duality between cutoff cyclones and blocking anticyclones ON THE CANCELLATION OF HORIZONTAL ADVECTION BY VERTICAL MOTION ANOMALIES AT THE LOWER BOUNDARY, AND THE INVERTIBILITY PRINCIPLE FOR GENERAL, TIME-DEPENDENT FLOW S(a) Surface and near-surface anomalies S(b) Quasi-geostrophic theory S(c) Semi-geostrophic theory and Salmon’s generalization 5(d) Inversion by nonlinear normal-mode initialization 6(a) Rossby wave propagation and the scale effect 6( b) Baroclinic and barotropic shear instabilities 6(c) Lateral and vertical Rossby wave propagation 6(d) The nonlinear saturation of baroclinic instabilities 6(e) Further remarks about cyclogenesis in the real atmosphere THE MAINTENANCE AND DISSIPATION OF CUTOFF CYCLONES AND BLOCKING FURTHER REMARKS ABOUT CUTOFF SYSTEMS AND AIR MASSES CONCLUDING REMARKS ROSSBY WAVES AND SHEAR INSTABILITIES ANTICYCLONES ACKNOWLEDGEMENTS APPENDIX: THE COMPUTATION OF VERTICAL MOTION 1. INTRODUCTION AND HISTORICAL REVIEW (a) Early ideas Circulation and vorticity have been recognized as fundamental concepts in meteor- ology and oceanography for many years, dating back to the pioneering work of V. Bjerknes (1898a, b, 1901, 1902); see also, e.g., Eliassen (1982). The three-dimensional vorticity equation, as it appears in textbooks on general fluid dynamics, may be written for frictionless motion relative to a coordinate system rotating with angular velocity Ln as where the absolute vorticity and the relative vorticity. D/Dt is the material rate of change, and V is the three-dimensional gradient operator. We denote the three-dimensional velocity by u = (u, u, w) to dis- tinguish it from the horizontal wind velocity v = (u, v,O). The first term on the right- hand side of (1) is the stretching-twisting term, and the second the so-called solenoid term. 5, =2ll+( (2) J=VXu, (3) The absolute circulation around a material circuit r moving with the fluid is C, = C -+ 2QA, (4)ISENTROPIC POTENTIAL VORTICITY MAPS 879 where C = fr u. dl (5) is the relative circulation and A is the area bounded by a projection of the