Animation of horizontally homogeneous non-rotating radiative-convective equilibrium courtesy of Caroline Muller. The model is SAM (System for Atmospheric Modeling) the principal architect of which is Marat Khairoutdinov. Transparent shading is condensate concentration; colors on the surface indicate near-surface air temperature. See text for further description.
The starting point for most of my thinking regarding climate sensitivity is the simple 1-dimensional radiative-convective model introduced by Suki Manabe and Dick Wetherald in 1967. See also Manabe and Strickler, 1964. For an early review of this kind of modeling, see Ramanathan and Coakley, 1978. Sadly, Dick Wetherald passed away very recently; although it is a very small gesture, I would like to dedicate this post to his memory.
This model solves for a single vertical temperature profile as a function of pressure (or height ) which one can think of as representing the globally average temperature profile. There is no explicit atmospheric circulation, although it is present implicitly. The atmosphere is in hydrostatic balance: . The model consists conceptually of two parts. There is a radiative transfer module that generates the net shortwave and longwave radiative fluxes, the sum of which I will call . As input to this module one needs the incident solar flux, surface albedo, and the vertical profiles of temperature and whatever radiatively active constituents are assumed present. Secondly, there is a convective flux that redistributes energy in the vertical:
The surface temperature is determined by the net surface radiation and the convective flux at the surface:
Here is the heat capacity at fixed pressure, is the density, and and are positive upwards. I have also given the surface some heat capacity — this value is not important for the steady state.
If you integrate such a model to equilibrium without any convective redistribution of energy — ie if you compute pure radiative equilibrium — the result will be strongly gravitationally unstable near the surface. A minimal model of convection might redistribute energy vertically whenever the lapse rate, , increases beyond the dry adiabatic value for an ideal gas, so as to bring the lapse rate back to this critical value. The mixing is assumed to be very strong once the critical lapse rate is reached — the assumption is that the time scales of the convective mixing are much smaller than those of the radiative relaxation towards equilibrium. You can formulate a mixing process that achieves this if you diffuse , the dry static energy, and turn on the diffusion only when the lapse rate exceeds this critical value. One also needs to model the surface convective flux. The simplest possible model is just to assume that the surface flux is zero except when it is needed to keep the surface from getting warmer than the surface air — ie, ignoring the air-surface temperature difference.
MW don’t actually adjust to the dry adiabatic value. If one does that, the tropopause is too low and the tropospheric lapse rate too large. The observed globally averaged lapse rate is about 6.5K/km. MW simply use this observed value for the critical lapse rate, which results in a reasonable tropopause height. Theories for the observed tropospheric lapse are not easily incorporated in this globally averaged framework, since different mechanisms stabilizing the troposphere are at work in the tropics and in higher latitudes. (In low latitudes, it is more natural to talk about a moist rather than the dry adiabatic lapse rate; in higher latitudes, the large-scale quasi-horizontal turbulence that produces highs and lows and weather competes with smaller scale moist convection in transporting energy upwards.)
In addition, MW do not bother with an explicit diffusive model for the convective transport. Instead they use a simple convective adjustment – while integrating towards equilibrium with a vertically finite-differenced model, check at every time step to see if the flow is unstable according to the prescribed critical lapse rate — if any two layers are unstable set the lapse rate between these two layers equal to the critical value while conserving the mean energy (here this is just the mean temperature) of the two layers. Do this also at the surface to prevent the first atmospheric layer immediately above the surface from being colder than the surface.
The tropopause height is part of the solution. The result for realistic settings is just a troposphere at the critical lapse rate merging continuously at the tropopause into a stratosphere in radiative equilibrium. If you know that this is what the equilibrium looks like, you can get the equilibrium solution by a simpler iteration. For a given surface temperature and tropopause height, the tropospheric temperatures are known. Given these temperatures you can compute radiative equilibrium above this tropopause. The solution will have two problems: the temperature will not be continuous at the tropopause, and the energy flux at the top of the atmosphere will not be zero. These two constraints can then be used to determined the two unknowns — the surface temperature and the tropopause height.
You can do more elaborate things with the surface fluxes and try to simulate the effective air-surface temperature difference, especially if you want to divide the surface convective flux into its two components, evaporation and sensible heat, but this extension doesn’t change the model’s climate sensitivity appreciably.
MW compare the assumption of fixing the relative humidity distribution in the troposphere to that of fixing specific humidities, providing the first modern estimates of the difference this makes for climate sensitivity. Stratospheric water is specified as is the ozone distribution. Clouds must also be prescribed in this model. Increasing the CO2 the surface and troposphere warm by the same amount, by construction, while the stratosphere cools and the tropopause rises, as described in MW. Is this very strong coupling of the troposphere to the surface realistic? I think it is a very good place to start, but my purpose in this post is not to convince you of that but just to convey what this radiative-convective model is.
The strong coupling requires one to think about the energy balance of the surface + troposphere rather than the surface in isolation. Suppose one puts a layer into the troposphere that absorbs some of the solar radiation without increasing the reflection. From a surface energy balance perspective one might guess that this would cool the surface, since less solar radiation would penetrate to the ground. But from the perspective of a strongly coupled surface-troposphere system, whether one absorbs at the surface or in the interior of the troposphere is irrelevant for the temperature response to first order — in fact this absorption would cause warming to the extent that it prevents the scattering to space that would otherwise occur (you maximize this effect by putting the absorber over ice or a low cloud deck.) It is interesting to ask how strong the absorption in the troposphere must be to decrease the convective mixing to the point that the surface decouples from the troposphere. We might call this the “nuclear winter” problem.
In the past one or two decades, there has been an increasing amount of work on radiative-convective models with explicit moist convection. Take your numerical model of the atmosphere and place it over a flat homogeneous surface, ignore rotation, and assume that the geometry is re-entrant in both horizontal dimensions. There are no walls and every point in the horizontal is physically identical to every other point. Assume that the surface is saturated — ie ocean. Turn on the radiative transfer and start destabilizing the atmosphere, evaporating water and generating cumulus convection. Its the same idea as the single column model, but now the model is determining its own clouds and water vapor distribution as well as temperature profile. (Typically one still fixes ozone and stratospheric water). The upper part of the animation above, kindly provided by Caroline Muller, has horizontal resolution of 2km and a square 200 x 200 km domain. This is a statistically steady state achieved after a couple of months of integration. See Tompkins and Craig, 1998 to read more about this kind of simulation. Romps, 2010 is a recent attempt to push to much higher horizontal resolution, to better resolve the key patterns of entrainment and detrainment into and out of the turbulent convective plumes.
The temperature profiles these models produce are qualitatively similar to those generated by single column convective adjustment models, with the moist adiabat determining the critical lapse rate. The surface and troposphere are very strongly coupled in these simulations. I’ll discuss the changes in water vapor and clouds that they simulate in response to CO2 in future posts.
You can get a taste for how these “cloud-resolving models” are compared to data from a variety of observational field programs here. We cannot test them in this homogeneous configuration — you naturally have to simulate the conditions in particular regions in which there have been field programs that provide appropriate data.
The lower panel in the animation at the top of the page is strikingly different from the upper panel, yet it is generated by simply increasing the size of the domain to 512 x 512 km. The convection now aggregates into a small fraction of the domain. See Bretherton et al 2005 for a discussion of this behavior. Caroline and I are currently re-examining theories of this self-aggregation in homogeneous models. The model has hysteresis for some parameter settings, so its climate is not always unique. I find this sort of thing challenging but frustrating as well. We saw something like this in an early low resolution 2-dimensional (x-z) study (Held et al, 1993), but I was hoping that the 3D case would be free of this kind of complexity, so that we could more easily use it as a stepping stone towards understanding more realistic models. Is self-aggregation in the statistically-steady homogeneously-forced non-rotating model a curiosity, or is it telling us something important?
[The views expressed on this blog are in no sense official positions of the Geophysical Fluid Dynamics Laboratory, the National Oceanic and Atmospheric Administration, or the Department of Commerce.]