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

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.

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

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.

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.

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

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)?

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, 361-385.

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.

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

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.

RCE Calculations with Emanuel single-column model (Renno et al RCE Calculations with Emanuel single-column model (Renno et al. 1994, JGR, 99, 14429-14441) 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

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!

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.

Example: simulations of ASTEX Lagrangian field experiment (1992) - Boundary layer dynamics Bretherton et al., Bound. Layer Meteor., 93, 341-380. 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.

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

Example: testing deep convection schemes Temperature error with (above) & without (below) downdrafts time Sud and Walker 1993, Mon. Wea. Rev. 121, 3019-3039 Adding downdrafts to a scheme, see the difference. Forced by time series of advection terms from GATE field experiment. Compare temperature & humidity to obs.

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)

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.

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.

Some WTG simulations (Sobel & Bretherton 2000, J Some WTG simulations (Sobel & Bretherton 2000, J. Climate 13, 4378-4392) 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, 1741-1766) to WTG RCMs with same physics, at each grid point

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

Temperature

Specific humidity

Relative humidity

Moist static energy