1. Kushner: GCC Lecture Topics 3-D Dynamical Models … ßTropospheric eddies and the general circulation. ßSimple troposphere and stratosphere-troposphere atmospheric general circulation models. … Climate Models (Tropospheric GCMs) … ßBuilding them and using them. … and a Little Philosophy ßPlease indulge me.

2. Three-Dimensional Dynamical Models, 1 Paul J. Kushner University of Toronto GCC Summer School Banff 2005

3. Outline Introduction: A little philosophy … Phenomenology: Large-scale waves and the atmospheric general circulation. Theory and simple models of transient eddies: baroclinic instabilities and lifecycles Introducing simple atmospheric general circulation models. Conclusion: Points to take hiking

4. Introduction: A Little Philosophy … Thanks to Isaac Held at NOAA/GFDL. We don’t want to get hung up on generalities, but it’s good to know what we’re doing and why we’re doing it. Conceptual models/pictures are useful but they are often ambiguous, only vaguely understood, and overplayed.

5. Some Common Conceptual Models Conveyor belt IPCC Radiative Forcing Graphic STE

6. In Defense of Simple Systems In addition to conceptual pictures, we need to understand simple physically realizable analog systems. Held (2005): ßBiologists agree without apology that a relentless focus on E. Coli, flies and mice will lead to progress in human biology. We should focus on studying such analogs in climate systems.

7. Today I will stress the importance of midlatitude weather (baroclinic eddies) to understand the tropospheric and stratospheric general circulation. I will try (not always successfully) to build my arguments on physically realizable systems.

8. Radiative Imbalance and Transport Top-of-Atmosphere Radiation Hartmann 1994 At each latitude, there is an imbalance between absorbed and emitted radiation This radiative imbalance can occur because the atmosphere and oceans can move energy around.

9. A Steady-State Model Imagine heating up water in a box. When heating and cooling are vertically distributed, various details determine the horizontal scales of motion. But with a lateral heating distribution, the circulation cell tends to fill the cell. In steady state, there is a balance between baroclinic generation of circulation and friction. Heat Source Heat Sink (Wallace and Hobbes)

10. A Steady-State Model But with a lateral heating distribution, the circulation cell tends to fill the cell. In steady state, there is a balance between baroclinic generation of circulation and friction. (Wallace and Hobbes)

11. An Adjustment Model Annual-mean Potential Temperature Adjustment of a baroclinic fluid Gill 1992 IRI/LDEO via J. Marshall Now suppose we turned off the heating. The fluid spontaneously adjusts to a stable equilibrium. The adjustment involves a “poleward” flux of heat and constituents. The distribution of potential temperature in the troposphere is analogous to these models. As of today, we still lack a theory for the slope of the isentropes in the troposphere.

12. Rotation and Jets are Important IRI/LDEO via J. Marshall Annual-mean Potential Temperature Annual-mean Zonal Wind The fluid-cell examples, to be relevant, need to be complemented by rotation. The reason is that for a rapidly rotating hydrostatic fluid, baroclinicity also gives rise to zonal jets. These jets in turn give rise to waves that can transport heat poleward much more efficiently than zonal motions.

13. Breakdown of Energy Transports Meridional Flux of Sensible Heat Peixoto and Oort 1992 Sensible-heat (T) transport is representative of the total energy transport. Zonally symmetric motions do most of the transport in the tropics. Waves do most of the transport in midlatitudes. Stationary waves are important in NH winter; At other times, and in the SH, transient waves dominate.

14. Laboratory Baroclinic Waves The rotating annulus below is heated on the inside ring (orange = warm, blue = cold) The flow remains annular until the rotation is increased enough. Then baroclinic instability sets in (right). http://dennou-k.gaia.h.kyoto-u.ac.jp/library/gfd_exp/exp_e/doc/bc/images/bc05.gif

15. Observed Wave Spectra Aircraft measurements of spatial spectra (upper trop) Gage and Nastrom 1985 The energetic waves have scales between 10 3 -10 4 km. The figure suggests a self-similar turbulent character. But in many ways these waves have a linear or weakly nonlinear character. We will focus on the transient waves but show stationary waves for comparison … animations

16. Midlatitude Baroclinic Waves Meridional Velocity Correlations Chang and Yu 1999, Chang 1993 These waves propagate in packets. Their group speed (blue) is greater than their phase speed (red). Their “crescent” shape implies a meridional convergence of zonal momentum. Their westward phase tilt with height implies a poleward heat transport.

17. Explanation of Phase Tilts Consider fluxes of momentum and of heat (for simplicity, flux of buoyancy). –

18. Midlatitude Stationary Waves DJF Z*500, NCEP Reanalysis DJF Z*(60N), NCEP Reanalysis Like the transient eddies, these waves are associated with momentum flux converges and poleward heat transports. Unlike the transient eddies, these waves amplify into the stratosphere.

19. Transient and Stationary Eddies Randel 1992 We see that both transient and stationary eddies play a large role in stratospheric variability and transports of heat and momentum.

20. On Eddy Fluxes of Heat and Momentum The stationary and transient eddies share some common features and their flux contributions rarely seem to cancel. We expect this reinforcement because of the robust tendency for atmospheric waves to flux potential vorticity downgradient. Dylan discussed this in the context of flux- gradient K yy formulations for the BDC. We’ll return to these ideas later.

21. Transient and Stationary Wave Models We will now run through a set of models that capture the main features of transient eddies, representing work that extends from the 1940’s until today. These all represent transient “initial-value problem” models where initial conditions matter. ßFor climate (equilibrated) models, initial conditions should be forgotten.

22. Initial Value Problems The first set of problems we’ll look at are linear wave problems. 1.We start with the primitive equations. 2.We scale them using the assumptions of quasigeostrophic scaling to obtain QG equations. 3.We then assume some basic flow and linearize about it (as Charles did). 4.Fast growing (unstable) waves are taken to be the relevant ones.

23. Models of Baroclinic Eddies: Phillips We linearize the equations about this basic state using QG scaling. When we add waves to this basic state, we find:

24. Holton: A Growing Baroclinic Wave This model has some of the basic elements we are looking for, in particular the westward phase tilt with height and the right kind of growth rates. Notice the x-z plane circulation accompanying the growing wave: all these agree qualitatively with observed waves.

25. Models of Baroclinic Eddies: Eady The fastest growing wave: 4000 km (m=6) Growth rate corresponds to about 2-3 days e- folding time:

26. Gill: A Growing Eady Wave This model has the appropriate phase tilt with height. It suggests that a strong interaction between the surface and tropopause can help baroclinic waves grow.

27. Models of Baroclinic Eddies: Charney Growth rate as in Eady model:

28. Gill: A Growing Charney Wave In this model, the vertical phase tilt and heat fluxes are more surface trapped. The vertical scale of the wave is a strong function of beta and u.

29. Linear Baroclinic Eddies: Simmons and Hoskins Basic State Wind, Temperature, and Growing Wave Simmons and Hoskins 1976 Modern computers made it possible to avoid QG scaling, include spherical effects and a realistic basic state. The resulting waves looked more and more like Chang’s regression maps. Notice how the unstable waves flux momentum into the jet core

30. Nonlinear Baroclinic Eddies Simmons and Hoskins 1978 Polvani et al. 2004 Nonlinear simulations allowed the baroclinic waves to run through life cycles of growth, saturation and decay. Notice how the waves sharpen the jet. The Polvani et al. calculations showed the packet structure clearly.

31. Transient Eddies in Climate Models

32. We now look for the simplest models that can maintain a general circulation similar to nature. We think of a dry atmosphere in which the temperature is relaxed back to a baroclinically unstable profile of temperature. The PBL and sponge (!) are damped with Rayleigh friction. This system is not fully realizable (Rayleigh drag). Animations …

33. Points to Take Hiking Baroclinic eddies play a central part in the general circulation, so we should get to know them. Baroclinic eddies can be captured by a sequence of more and more sophisticated models. By putting them in a climate setting (the simple AGCM), we will now be able to see how baroclinic eddies: ßSet the stratification of the troposphere. ßInfluence the connection of the stratosphere to the troposphere.