Author's personal copy Provided for non-commercial research and educational use only. Not for reproduction, distribution or commercial use. This article was originally published in the book Encyclopedia of Atmospheric Sciences, 2nd edition. The copy attached is provided by Elsevier for the author's benefit and for the benefit of the author's institution, for non-commercial research, and educational use. This includes without limitation use in instruction at your institution, distribution to specific colleagues, and providing a copy to your institution's administrator. All other uses, reproduction and distribution, including without limitation commercial reprints, selling or licensing copies or access, or posting on open internet sites, your personal or institution’s website or repository, are prohibited. For exceptions, permission may be sought for such use through Elsevier’s permissions site at: http://www.elsevier.com/locate/permissionusematerial From McIntyre, M.E., 2015. Balanced Flow. In: Gerald R. North (editor-in-chief), John Pyle and Fuqing Zhang (editors). Encyclopedia of Atmospheric Sciences, 2nd edition, Vol 2, pp. 298–303. ISBN: 9780123822253 Copyright © 2015 Elsevier Ltd. unless otherwise stated. All rights reserved. Academic Press Author's personal copy Balanced Flow ME McIntyre, University of Cambridge, Cambridge, UK Ó 2015 Elsevier Ltd. All rights reserved. Synopsis A balanced flow is one in which the three-dimensional velocity field is functionally related to the mass field, presumed hydrostatically related to the pressure field. Such a functional relation between the velocity and mass fields is called a balance relation, or filtering condition. The simplest but least accurate such relation is geostrophic balance. There are more accurate balance relations, of which the most accurate are fully nonlocal. That is, the velocity at a point depends on the mass field throughout the domain. There are ultimate limitations to accuracy, governed by the fuzziness of the slow quasimanifold. Introduction (x6ofDynamical Meteorology: Potential Vorticity). In these theories the mean or vortical flow is usually considered to be The concept of balanced flow is the counterpart, in atmosphere– balanced, regardless of the wave types involved. Indeed, the ocean dynamics, to the well-known concept of nearly incom- concept of balance enters, implicitly or explicitly, into almost pressible flow in classical aerodynamics. In aerodynamics, a key any discussion of meteorologically interesting fluid aspect of such flow – long recognized as central to under- phenomena; and balance versus imbalance is part of the standing the behavior of subsonic aircraft – is that all the conceptual foundation that underpins data analysis, data significant dynamical information is contained in the vorticity assimilation, and weather prediction. field. To that extent the flow has, in effect, fewer degrees of freedom than a fully general flow. We may think of it as being elastostatically balanced, in the sense that freely propagating The Elastic Pendulum sound waves can be neglected in the dynamics. In atmosphere–ocean dynamics there is a corresponding Balance has counterparts not only in aerodynamics but also in statement with vorticity replaced by potential vorticity (PV), simple mechanical systems such as the elastic pendulum. This is understood in a suitably generalized sense; see generalized PV a massive bob suspended from a pivot by a stiff elastic spring of field in the article Dynamical Meteorology: Potential Vorticity. negligible mass. Such a pendulum has slow, swinging modes of For many cases of rotating, stably stratified fluid flow, with oscillation in which the relatively fast, compressional modes of parameter values typical of the atmosphere and oceans, all the the bob and spring are hardly excited: they can be neglected in significant dynamical information is contained in the general- the dynamics if the spring is stiff enough. The slow, swinging ized PV field. One may invert this field at each instant to obtain modes correspond to balanced flow, and the fast, compres- the mass and velocity fields. The article on Potential Vorticity sional modes to sound and inertia–gravity waves. One may gives a more precise statement. All such flows may be charac- describe the swinging modes to a crude first approximation by terized as balanced. making the spring strictly incompressible, i.e., by making its Again this means that the flow has, in effect, fewer degrees of length strictly constant. There is a hierarchy of more accurate freedom than a fully general flow. More precisely, balance and approximations that allow the spring to change its length in invertibility mean that not only sound waves but also freely a quasi-static or elastostatic way, the spring being longest when propagating inertia–gravity waves can be neglected in, or filtered the bob moves fastest and shortest when the bob is stationary. from, the dynamics. Thus balanced flows can be much simpler In such a quasi-static description the length of the spring is to understand than fully general flows, thanks to the relatively functionally related to the speed of the bob. The functional simple way in which the advective nonlinearity acts on the PV. relation holds at each instant t, i.e., it holds diagnostically. No Cases of fluid flow describable as balanced come under derivatives or integrals with respect to t are involved, and values headings such as Rossby waves, Rossby-wave breaking, vortex of t do not explicitly enter into the definition of the functional dynamics, vortical modes, vortical flow, vortex coherence, relation. The property of being diagnostic, in this sense, vortex resilience, eddy-transport barriers, blocking, cyclogen- provides us with a useful mathematical and conceptual esis, baroclinic instability and barotropic instability (meaning simplification. the wavy shear instabilities), all of which are related to the Such approximations and their ultimate limitations can be fundamental Rossby-wave restoring mechanism or quasi- studied mathematically via techniques ranging all the way elasticity that exists whenever there are isentropic gradients of from two-timing formalisms (method of multiple scales) and PV in the interior of the flow domain, or gradients of potential bounded-derivative theory to KAM (Kolmogorov–Arnol’d– temperature on an upper or lower boundary. The concept of Moser) theory and other dynamical-systems techniques; balanced flow is fundamental, also, to theories of wave–mean there is an enormous literature. interaction and wave–vortex interaction, needed in order to The error incurred in using the most accurate quasi-static understand, for instance, the gyroscopic pumping that drives descriptions becomes exponentially small as the fast–slow global-scale stratospheric circulations and chemical transports timescale separation increases. It may even be zero, or in some 298 Encyclopedia of Atmospheric Sciences 2nd Edition, Volume 2 http://dx.doi.org/10.1016/B978-0-12-382225-3.00484-9 Encyclopedia of Atmospheric Sciences, Second Edition, 2015, 298–303 Author's personal copy Dynamical Meteorology j Balanced Flow 299 circumstances small but inherently nonzero (corresponding to acceleration), and position x is specified using pressure altitude KAM tori breaking into thin chaotic layers, also called fractal along with horizontal position x, y. Thus the horizontal spatial layers or stochastic layers). derivatives v/vx and v/vy are taken at constant pressure altitude rather than at constant geometric altitude. This qualifies as a balance relation because of the presumption that the Balance Relations hydrostatic relation also holds, as normally assumed when using pressure as the vertical coordinate. Knowing F on each In atmosphere–ocean dynamics the defining property of balance constant-pressure (isobaric) surface is then equivalent to is that an analogous functional relation holds – diagnostic in knowing the mass field. So eqn [1] is, as required, a diagnostic precisely the same sense. The functional relation between bob functional relation between the velocity field and the mass speed and spring length is replaced by a functional relation field. The vertical derivative of eqn [1] is the so-called thermal between the fluid’s velocity and mass fields. More precisely, wind equation. a flow is said to be balanced if the three-dimensional velocity The horizontal coordinates x, y are orthogonal coordinates, field u(x,t) satisfies a functional relation of the form u(x,t) ¼ uB and can be taken either as local curvilinear following the Earth’s where uB depends only on the mass field or mass configuration, geometry, or as local Cartesian in a tangent-plane approxima- i.e., on the spatial distribution of mass throughout the fluid tion. If we also take f ¼ constant, giving us the so-called f-plane system, presumed to be hydrostatically related to the pressure approximation, then eqn [1] asserts not only that u is slaved to field. (Knowledge of the mass field then implies knowledge of the mass field but also that it is two-dimensionally incom- the pressure, temperature and potential temperature fields, given pressible or nondivergent, with streamfunction J ¼ F/f, so that zero pressure at the top of the atmosphere.) vJ vJ Such a functional relation u(x,t) ¼ uB between the velocity ð ; Þ¼ À ; ; u x t v v 0 [2] and mass fields is called a filtering or balance condition, or y x balance relation. It supplies just enough information to make The geostrophic relation [1] – or relations, plural, if one the PV field invertible. The property