## GREB Decomposition: Hydrology

Conceptually, the hydrology is a very simple model. Water vapor evaporation is driven by surface temperature, wind speed, soil moisture, and surface height via an extended bulk formula. Precipitation is proportional to the water vapor in the atmosphere. I’ve annotated the GREB code below to reference the equations as listed in Dommenget and Floeter (2011).

3.3 Hydrological cycleThe response in viwv

_{atmos}and the latent heat release is central to climate change. We therefore have to simulate the response of the hydrological cycle, which includes the evaporation of water vapor at the surface, the condensation of water in the atmosphere and the associated take-up and releases of latent heat at the surface and in the atmosphere, respectively. The saturation surface air layer specific humidity, q_{sat}, is given by

q_{sat}= e^{-ztopo/zatmos}* 3.75*10^{-3}* e^{( 17:08085*(Tsurf-273.15)/(Tsurf-273.15) )}(6)which is taken from the textbook from James (1994) and extended to consider changes in surface pressure due to the topographic height, z

_{topo}. The latent heat release to the surface layer associated with evaporation is given by an extended Bulk formula (Peixoto and Oort 1992a, 1992b):F

_{latent}= L * rho_{air}* C_{w}* abs(u) * theta_{soil}* (q_{air}– q_{sat}) (7)The Bulk formula depends on the difference between q

_{sat}and the actual surface air layer humidity, q_{air}, the wind speed, abs(u) the constant parameters of the latent heat of evaporation and condensation of water, L, the density of air, q_{air}, rho_{air}and the transfer coefficient, C_{w}. The wind speed, abs(u) is assumed to be the seasonally varying mean winds of the NCEP reanalysis 850 hPa geopotential height winds, abs(u) (see Fig. 1e for the annual mean values) plus a globally constant turbulent part of 3 m/s over oceans and 2 m/s over land. The Bulk formula is extended by a surface wetness fraction, theta_{soil}, to simulate evaporation over land, where the surface is not always wet. theta_{soil}is assumed to be a climatological boundary condition in the GREB model, which however varies with the seasonal cycle, see Fig. 1g for the annual mean values.The atmospheric integrated water vapor, viwv

_{atmos}, is roughly linearly related to the near surface humidity q_{air}(e.g. Rapti 2005), which is estimated by a linear regression from ECHAM5 simulations scaled by topography:VIWV

_{atmos}= e^{-ztopo/zatmos}* 2.6736*10^{3}[kg/m^{2}] * q_{air}(8)Note, that the additional scaling by the topography should simulate the effect of nearly exponentially decreasing atmospheric water vapor mixing ratios. Changes per unit time in q

_{air}by evaporation, dq_{eva}, are given with the help of Eq. 7:dq

_{eva}= (-F_{latent}/L) * 1 / 2.6736*10^{3}[kg/m^{2}] (9)The latent heat release, Q

_{latent}, in the atmosphere due to changes in q_{air}by condensation or precipitation, dq_{precip}, is given byQ

_{latent}= – 2.6736*10^{3}[kg/m^{2}] * dq_{precip}* L (10)The condensation or precipitation, dq

_{precip}is assumed to be proportional to q_{air}dq

_{precip}= r_{precip}* q_{air}(11)with r

_{precip}= -0.1/24 h, which corresponds to an autoregressive model with a decorrelation (recirculation)

Equation 6 is the Clausius-Clapeyron equation.

The transition from eq 7 to eq 9 was not obvious to me until I found this relationship in the teaching material.

∆q_{eva} = 1 / r_{H2O} · ρair · Cw · |u∗ | · v_{soil} · (q_{sat} − q_{atmos} ) (4.13)

Precipitation is given by equation 11 and which sets the precipitation at 10% of the water vapor per day.

And now, the code …

!+++++++++++++++++++++++++++++++++++++++ subroutine hydro(Tsurf, q, Qlat, Qlat_air, dq_eva, dq_rain) !+++++++++++++++++++++++++++++++++++++++ ! hydrological model for latent heat and water vapor USE mo_numerics, ONLY: xdim, ydim USE mo_physics, ONLY: rho_air, uclim, vclim, z_topo, swetclim, ityr, & & ce, cq_latent, cq_rain, z_air, r_qviwv, log_exp ! declare temporary fields real, dimension(xdim,ydim) :: Tsurf, q, Qlat, Qlat_air, qs, dq_eva, & & dq_rain, abswind Qlat=0.; Qlat_air=0.; dq_eva=0.; dq_rain=0. if(log_exp 0. ) abswind = sqrt(abswind**2 +2.0**2) ! land where(z_topo < 0. ) abswind = sqrt(abswind**2 +3.0**2) ! ocean ! saturated humiditiy (max. air water vapor) qs = 3.75e-3*exp(17.08085*(Tsurf-273.15)/(Tsurf-38.975)); ! eq (6) qs = qs*exp(-z_topo/z_air) ! scale qs by topography ! eq (6) ! latent heat flux surface Qlat = (q-qs)*abswind*cq_latent*rho_air*ce*swetclim(:,:,ityr) ! eq (7) ! change in water vapor dq_eva = -Qlat/cq_latent/r_qviwv ! evaporation ! eq(9) ! cq_latent = 2.257e6 dq_rain = cq_rain*q ! rain ! eq(11) !cq_rain = -0.1/24./3600 ! latent heat flux atmos Qlat_air = -dq_rain*cq_latent*r_qviwv ! eq (10) end subroutine hydro

Wiki states: *Again assuming constant relative humidity, the Clausius-Clapeyron equation shows that water vapour increases roughly exponentially with temperature, at approximately 7% for typical temperatures.*

The model responds appropriately when I increase surface temperatures by increasing solar insolation – rising 6-7% per degC increase in surface temperatures. Interestingly, the model’s water vapor response curve is slightly less when the surface temp is raised by CO2 forcing. The results are displayed in the chart above.

—————–

Dommenget, D., and J. Floeter 2011: Conceptual Understanding of Climate Change with a Globally Resolved Energy Balance Model. Climate dynamics, 2011, 37, 2143-2165.

## GREB Decomposition: Long Wave Radiation

Diving once again into Conceptual understanding of climate change with a globally

resolved energy balance model (Dommenget and Floter, 2011), this time to flesh out the emissivity and long wave radiation within the model.

## GREB Decomposition: Solar Radiation

I turn to Dommenget and Floter’s 2011Conceptual understanding of climate change with a globally

resolved energy balance model for description of the treatment of solar radiation and albedo within the model.I have reformatted some equations and symbols for readability.

## GREB: Multilevel CO2 Doubling Experiment

Now that we’ve got a handle on the input and output files, let’s take the GREB model out for a test drive.

## GREB: Globally Resolved Energy Balance

http://users.monash.edu.au/~dietmard/content/GREB/GREB_model.html

Eli points to the Monash Simple Climate Model which is “based” on the Globally Resolved Energy Balance (GREB) published by Dommenget and Floeter (2011) in *Climate Dyanmics*. I’m not sure if there are any difference between the models beyond the name. It appears that the Monash Simple Climate Model is an attempt to build a climate education site using comparisons between canned scenarios to teach the basic components in climate dynamics. I found the website a bit clunky. However, by following the links, you can find code.

## van Hateren 2012: May the Schwartz be with you

Figure 6. Isomorphism between resistance-capacitance circuit and two-compartment energy balance climate model. Differential equations on right can be solved to give time dependence for arbitrary applied time-dependent forcing (current). Dashed boxes enclose corresponding one-compartment systems. The figure is modified from the Reply to Comments on my 2007 paper and an in-press paper (Spring, 2012) that interprets the observed increase in GMST over the latter part of the twentieth century in terms of the two-compartment model.

Stephen E. Schwartz Home Page

http://www.ecd.bnl.gov/steve/schwartz.html

## UVic_ESCM 2.8 Build Notes

*A new earth system climate model of intermediate complexity has been developed and its climatology compared to observations. The UVic Earth System Climate Model consists of a three-dimensional ocean general circulation model coupled to a thermodynamic/dynamic sea-ice model, an energy-moisture balance atmospheric model with dynamical feedbacks, and a thermomechanical land-ice model. In order to keep the model computationally efficient a reduced complexity atmosphere model is used. Atmospheric heat and freshwater transports are parametrized through Fickian diffusion, and precipitation is assumed to occur when the relative humidity is greater than 85%. Moisture transport can also be accomplished through advection if desired. Precipitation over land is assumed to return instantaneously to the ocean via one of 33 observed river drainage basins. Ice and snow albedo feedbacks are included in the coupled model by locally increasing the prescribed latitudinal profile of the planetary albedo. The atmospheric model includes a parametrization of water vapour/planetary longwave feedbacks, although the radiative forcing associated with changes in atmospheric CO2 is prescribed as a modification of the planetary longwave radiative flux. A specified lapse rate is used to reduce the surface temperature over land where there is topography. The model uses prescribed present-day winds in its climatology, although a dynamical wind feedback is included which exploits a latitudinally-varying empirical relationship between atmospheric surface temperature and density. The ocean component of the coupled model is based on the Geophysical Fluid Dynamics Laboratory (GFDL) Modular Ocean Model 2.2, with a global resolution of 3.6° (zonal) by 1.8° (meridional) and 19 vertical levels, and includes an option for brine-rejection parametrization. The sea-ice component incorporates an elastic-viscous-plastic rheology to represent sea-ice dynamics and various options for the representation of sea-ice thermodynamics and thickness distribution. The systematic comparison of the coupled model with observations reveals good agreement, especially when moisture transport is accomplished through advection.*

**The UVic Earth System Climate Model: Model Description, Climatology, and Applications to Past, Present and Future Climates**

Andrew J. Weaver, Michael Eby, Edward C. Wiebe, Cecilia M. Bitz, Phil B. Duffy,

Tracy L. Ewen, Augustus F. Fanning, Marika M. Holland, Amy MacFadyen, H. Damon Matthews,

Katrin J. Meissner, Oleg Saenko, Andreas Schmittner, Huaxiao Wang and Masakazu Yoshimori (2001)

http://mgg.coas.oregonstate.edu/~andreas/pdf/W/weaver01ao.pdf