1. Radiative-convective models
Adam Sobel
Columbia University
GCC Summer School, Banff

2. What is a radiative-convective (or, single-column) model?
Basically, a model which has no horizontal degrees
of freedom and so represents vertical energy
transfers only. It is a purely thermodynamic model;
cannot model momentum in any reasonable way
without a horizontal coordinate.

3. Fundamentals (e.g., Yanai et al. 1973, JAS 30, 611-627)
Start from primitive equations for dry static energy
s=cpT + gz and specific humidity q. Average over a
“grid cell” and make some standard assumptions, and
obtain equations for the grid-averaged variables
Where the convective heating Qc = Lv(c-e)+ /p (‘s’)
(L_v = latent heat of condensation, c=condensation, e=evaporation)
And “moisture sink” Qq = (e-c) + /p (‘q’)
sQc dp = Lv P + ‘s’|ps = LvP + H
sQq dp = -P + ‘q’|ps = E-P
integrals taken from surface up to nominal tropopause where  = 0

4. Assumptions
Horizontal transfers of energy at the subgrid scale are negligible compared to those in the vertical - “plane-parallel” assumption for convective and turbulent fluxes as well as radiative (QR=FR/p)
We will need to parameterize the terms on RHS – Qc, QR, Qq, in terms of mean quantities T, q (same as in GCMs). Not at all obvious that this can be done.

5. What is a radiative-convective (or, single-column) model?
An RCM solves the equations for s (equivalently, T or )
and q :
At a single horizontal location, representing an area from
a GCM grid box to an entire planet. The “physics” terms,
Q’s, are parameterized. The advection terms, containing
u and , cannot be computed internally, as the velocities
depend on momentum, incl. pressure gradients etc.

6. What do we do about the advection terms? 3 choices:
Assume they are zero – appropriate for an entire planet. Then the steady state is “radiative-convective equilibrium”: Qc = QR, and E=P
Prescribe them – from observations, or some other way
Parameterize them too, somehow

7. Application 1a: global mean climate theory
A la Manabe and colleagues, 1960’s
Advection terms set to zero, as appropriate for global average
counterpart to “energy balance models” which include meridional transport, but whose vertical energy transfers are highly simplified; in RCM, vertical is taken more seriously, horizontal is not dealt with
questions: what determines the global mean temperature of the planet, at surface and in free troposphere? How do tropospheric temperature structure and absorber amounts influence middle atmosphere (assumed in radiative equilibrium)?

8. Radiative vs. radiative-convective equilibria
Simple convective
scheme: adjustment of
T(z) to prevent lapse rate
from exceeding critical value.
No explicit consideration of
humidity (except radiatively). Energy conserved.
Warms the troposphere and
cools the surface.
Manabe and Strickler 1964, J. Atmos. Sci. 21,

9. A brief word about convective parameterizations
Most modern convective schemes are, one way or
another, adjustment schemes. This means that they
tend to force the temperature profile towards a particular
vertical structure, usually something close to a moist
adiabat. The scheme fires if the near-surface air is
relatively warm and moist (high moist static energy or
equivalent potential temperature) compared to the
dry static energy or potential temperature of the free
troposphere. The scheme will then warm the troposphere
and/or cool and dry the boundary layer to stabilize
the sounding. Modern schemes do not require grid-scale
saturation in order to fire – though Manabe’s MCA did.

10. Water vapor feedback Based on obs of seasonal
cycle, assume fixed relative
rather than specific humidity
in simple RCM. Result: climate
sensitivity to changes in CO2,
etc., significantly increased.
Quantitatively, this result is
in the range obtained by full
coupled GCMs today.
Manabe and Wetherald, 1967: Thermal Equilibrium of the
Atmosphere with a Given Distribution of Relative Humidity
JAS, 24, 241–259

11. Example 1b: spatially varying RCE as an input to dry models for the circulation
Some features of the atmospheric circulation may be
understood in terms of dry dynamics. Models in which the
thermodynamic equation is
d/dt = (e(y,z) - )/
have been useful for this purpose (e.g., Schneider and
Lindzen 1977; Held and Hou 1980) Since for a steady solution
with no flow  = e, e is the RCE solution. The idea is that
the moist physics can be entirely parameterized as relaxing
towards e on a fixed time scale . To complete the theory,
we have to compute e with an explicit moist model.

12. RCE Calculations with Emanuel single-column model (Renno et al
RCE Calculations with Emanuel single-column model (Renno et al. 1994, JGR, 99, )
Convective and (clear-sky) radiation parameterizations
Slab ocean mixed layer
Annual average insolation (function of latitude)
CO2, Ozone at reasonable values
Surface wind speed=7 m/s
Surface albedo tuned to give reasonable climate near equator

13. Rad-conv. equilibrium temperature as a function of latitude, annual average radiation
Albedo tuned to give ~realistic sounding at equator. Note
x axis only goes to 45 degrees, already well below freezing
at surface. Circulation needed to warm up Canada!

14. Application 2: testing physical parameterizations
“test bed” for GCMs – see how the parameterizations function without added complexity of interactions with dynamics
Large-scale advection terms specified, e.g. from obs – though this is not trivial to do, as requires accurate divergence & vertical velocity. In most places routine obs are not enough, field expts required
Compare T and q to obs. Comparing precipitation or convective heating is not generally meaningful.

15. Example: simulations of ASTEX Lagrangian field experiment (1992) - Boundary layer dynamics
Bretherton et al., Bound. Layer Meteor., 93,
½ the models were actually “LES” rather than “SCM”
Questions: what controls low cloudiness? Entrainment
physics, cloud microphysics, surface flux feedbacks…
but just simulations of ABL, free troposphere essentially
completely given. Single column follows low-level flow.

16. liquid water path
inversion height
Rate of change of
inversion height is
a measure of entrainment
rate
Related to cloud albedo

17. Example: testing deep convection schemes
Temperature
error with (above)
& without (below)
downdrafts
time
Sud and Walker 1993, Mon. Wea. Rev. 121,
Adding downdrafts to a scheme, see the difference. Forced by time series of advection terms from GATE field experiment. Compare temperature & humidity to
obs.

18. Precipitation is not a meaningful variable to compare to obs!
Dominant balance in s
equation is s/p ¼ Qc + QR,
variations Qc‘À QR’, and s/p ¼ const,
so Qc», and P»sQc dp. Thus if  given,
P is too, almost independently of model physics
(which is what we’re trying to test)

19. Errors in T, q in such tests are more meaningful. But
even then, success doesn’t mean the scheme is “correct”,
because in the full GCM, the dynamics are different:
the scheme will be interacting with the advection terms.
In the deep tropics, it may be just as valid to consider the
rather different limit in which  is completely
determined by the heating; the s equation reduces to:
s/p = Qc + QR
which is diagnostic for \omega, instead of prognostic for
s (or T). We must give up on prediction of temperature
but that’s ok, because it doesn’t vary much anyway in the
deep tropics. This is the so-called “weak temperature
gradient” (WTG) approximation.

20. Justification for WTG Precip is a more interesting
field than tropospheric
temperature, in the tropics.
Tropospheric temperature
has very small gradients.
Precip looks something like
SST, but sharper.

21. Some WTG simulations (Sobel & Bretherton 2000, J
Some WTG simulations (Sobel & Bretherton 2000, J. Climate 13, )
Comparing obs SST-P
relation to that computed
using Emanuel RCM
with T(p) held fixed, no
u¢rq
Comparing precip in “QTCM”
of Neelin and Zeng (2000, JAS
57, )
to WTG RCMs with same
physics, at each grid point

22. Summary
An RCM is a 1D model representing vertical energy transfers – advective, radiative, convective
Advective terms must be imposed, ignored, or (in some recent approaches) parameterized
Advective terms = 0 is “radiative-convective equilibrium”, which is adequate to understand some gross aspects of the global mean climate
Advective terms imposed when representing a single column, e.g. for testing parameterizations
In the latter case, precipitation/heating is not a meaningful output (at least over tropical oceans)
There are some new approaches to parameterizing large-scale dynamics for tropical applications

23. Temperature

24. Specific humidity

25. Relative humidity

26. Moist static energy