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Baroclinic and Barotropic Instabilities in Planetary Atmospheres: Energetics, Equilibration and Adjustment

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Baroclinic and Barotropic Instabilities in Planetary Atmospheres: Energetics, Equilibration and Adjustment Nonlin. Processes Geophys., 27, 147–173, 2020 https://doi.org/10.5194/npg-27-147-2020 © Author(s) 2020. This work is distributed under the Creative Commons Attribution 4.0 License. Baroclinic and barotropic instabilities in planetary atmospheres: energetics, equilibration and adjustment Peter Read1,z, Daniel Kennedy2,1, Neil Lewis1, Hélène Scolan3,1, Fachreddin Tabataba-Vakili4,1, Yixiong Wang1, Susie Wright1, and Roland Young5,1 1Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, UK 2Max-Planck-Institut für Plasmaphysik, Greifswald, Germany 3Laboratoire de Mécanique des Fluides et d’Acoustique, Université de Lyon, France 4Jet Propulsion Laboratory, Pasadena, California, USA 5Department of Physics & National Space Science and Technology Center, United Arab Emirates University, Al Ain, United Arab Emirates zInvited contribution by Peter Read, recipient of the EGU Lewis Fry Richardson Medal 2016. Correspondence: Peter Read ([email protected]) Received: 3 October 2019 – Discussion started: 15 October 2019 Revised: 14 February 2020 – Accepted: 26 February 2020 – Published: 3 April 2020 Abstract. Baroclinic and barotropic instabilities are well stabilities may efficiently mix potential vorticity to result in known as the mechanisms responsible for the production of a flow configuration that is found to approach a marginally the dominant energy-containing eddies in the atmospheres of unstable state with respect to Arnol’d’s second stability theo- Earth and several other planets, as well as Earth’s oceans. rem. We discuss the implications of these findings and iden- Here we consider insights provided by both linear and non- tify some outstanding open questions. linear instability theories into the conditions under which such instabilities may occur, with reference to forced and dissipative flows obtainable in the laboratory, in simplified numerical atmospheric circulation models and in the plan- 1 Introduction ets of our solar system. The equilibration of such instabili- ties is also of great importance in understanding the structure One of the great achievements of the past 100 years in fluid and energetics of the observable circulation of atmospheres dynamics has been the development of a theory of dynam- and oceans. Various ideas have been proposed concerning the ical instability, both linear and nonlinear. In a geophysical ways in which baroclinic and barotropic instabilities grow to context, this has led to a quantitative understanding of a va- a large amplitude and saturate whilst also modifying their riety of phenomena, including the processes that lead to the background flow and environment. This remains an area that development of large-scale energetic eddies in rotating, strat- continues to challenge theoreticians and observers, though ified atmospheres and oceans. These instabilities, known as some progress has been made. The notion that such instabili- baroclinic and barotropic instabilities, are well known as the ties may act under some conditions to adjust the background mechanisms responsible for the production of the dominant flow towards a critical state is explored here in the context energy-containing eddies in the atmospheres of Earth and of both laboratory systems and planetary atmospheres. Ev- several other planets, as well as Earth’s oceans. In general idence for such adjustment processes is found relating to they occur in flows for which background rotation plays an baroclinic instabilities under a range of conditions where the important role, while baroclinic instabilities also require stat- efficiency of eddy and zonal-mean heat transport may mu- ically stable stratification. They are typically distinguished tually compensate in maintaining a nearly invariant thermal energetically by the respective dominance of exchanges of structure in the zonal mean. In other systems, barotropic in- either kinetic or potential energy with the background (usu- ally zonal) flow. Barotropic instability is associated with di- Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union. 148 P. Read et al.: Baroclinic and barotropic adjustment rect exchanges of kinetic energy between background flow and nonlinear development of such instabilities governs the and eddies, while baroclinic instability entails the growth of structure and energetics of their circulation is still very much eddy kinetic and potential energy at the primary expense of the subject of ongoing research, using both observations and the potential energy of the background flow. numerical circulation models. Many planetary atmospheric These exchanges of energy between zonally symmetric circulation systems, for example, manifest complex, often flows and eddies and between potential and kinetic ener- anisotropic and inhomogeneous macroturbulent cascades of gies are commonly quantified in the manner originally sug- energy and enstrophy (squared vorticity), but how are such gested by Lorenz(1955, 1967), building on the much earlier cascades energised, and what processes determine the typi- work of Margules(1903), in which the flow is partitioned cal equilibrated distribution of energy? How are such states between zonally symmetric (zonally averaged) components maintained by the input of thermal energy from the Sun or and departures therefrom (“eddies”), with four main energy from the deep planetary interior? reservoirs (AE, AZ, KE and KZ) representing respectively Determining the occurrence or otherwise of either the eddy and zonal available potential energy and the corre- barotropic or baroclinic instabilities has traditionally ap- sponding kinetic energies. From considerations of the energy pealed to the results of linearised theories, often (though conservation equations, expressions can be defined to repre- not exclusively) derived from simplified forms of the equa- sent the rates of conversion between the four energy reser- tions of motion, continuity and energy conservation. The voirs (e.g. see Lorenz, 1967; Peixoto and Oort, 1974; James, most well known approach derives from the Charney–Stern– 1994), typically shown schematically in Fig.1a. Pedlosky (CSP) criterion for the existence of a non-zero As will be discussed further below, baroclinic or growth rate of an infinitesimal perturbation to a background barotropic instabilities and axisymmetric overturning circu- zonal flow (e.g. Vallis, 2017). This essentially reduces to the lations can be distinguished energetically by differing routes requirement that of energy conversion between potential and kinetic energies. But schematically, barotropic instabilities are characterised y28 H 9 Z <Z @Q=@y 2 F @U=@z 2H = by the dominance of the CK conversion term from zonal- −c Q dz C Q dy D 0; i j − j2 j − j2 mean kinetic energy KZ to eddy KE (see Fig.1c), while en- : U cr U cr 0 ; y1 0 ergy conversions in baroclinic instabilities are dominated by (1) the CA and CE terms, converting zonal-mean potential en- ergy AZ to eddy potential and kinetic energy AE and KE where U.y;z/ is the background zonal flow, Q.y;z/ is the (see Fig.1d). The direct kinetic energy conversion between corresponding quasi-geostrophic potential vorticity (QGPV), KE and KZ may also take place in the latter case in either y is the lateral and z is the vertical coordinate, c D cr C ici direction and so is indicated by the dashed arrows in Fig.1d. is the complex phase speed of the perturbation, Q .y;z/ is Such circulations may also be accompanied by direct, ax- the amplitude of the perturbation streamfunction, and F D isymmetric overturning circulations that primarily involve f 2=N 2. f D 2sinφ is the mean Coriolis parameter, cen- the CZ conversion between AZ and KZ (e.g. see Fig.1b). 0 0 0 tred at latitude φ0 with the planetary rotation rate, and N In addition to straightforward energy considerations, how- is the Brunt–Väisälä frequency. The integrals are carried out ever, the vorticity configuration (either absolute or potential over a domain in .y;z/ bounded by rigid walls or other suit- vorticity) also plays a key role in determining whether an in- able boundary conditions at y D y ;y and z D 0;H . stability develops and the character of the instability when it 1 2 For ci 6D 0, therefore, we require one or more of the fol- occurs. The subsequent nonlinear development of the insta- lowing criteria to be satisfied: bility then depends also on both energetic and vorticity con- straints, posing significant challenges to our understanding (i) @Q=@y changes sign in the interior domain; of how such instabilities will equilibrate. Predicting and quantifying such equilibration processes (ii) the interior @Q=@y take the opposite sign to @U=@z at are crucial tasks if we are to gain a full understanding of the upper boundary at z D H; the factors that govern the observable structure and evolu- (iii) the interior @Q=@y take the same sign as @U=@z at the tion of atmospheric circulation systems under a variety of lower boundary at z D 0; or different conditions. Recent exploration of our solar system has already provided much information on the structure and (iv) @Q=@y D 0 in the interior, and @U=@z takes the same properties of large- and medium-scale circulation systems sign at both z D 0 and z D H . across a wide range of parameter space, from very slowly rotating terrestrial planets such as Venus and Titan through Thus, Earth’s atmosphere typically satisfies criterion (iii), be- rapidly rotating terrestrial planets such as Earth and Mars ing dominated by the planetary vorticity gradient β D df=dy to the rapidly rotating gas and ice giants. Baroclinic and/or away from the lower boundary and @U=@z > 0 at the ground, barotropic instabilities likely play important roles in most of consistent with geostrophic balance and an equatorward tem- these atmospheric systems, but precisely how the occurrence perature gradient. In the oceans, in contrast, either criteria (i) Nonlin. Processes Geophys., 27, 147–173, 2020 www.nonlin-processes-geophys.net/27/147/2020/ P. Read et al.: Baroclinic and barotropic adjustment 149 Figure 1.
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