## 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.

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