EMAC - 3D MODEL 79

10000 100000 1e+06 1e+07 1e+08

350 400 450 500 550 600 650

pvap(Pa)

T(K)

Wagner and Pru\ss 2002 data table 1D model steam table Sonntag 1994 expression Wagner and Pru\ss 2002 expression

Figure 5.2: Saturation water vapor pressures formulations and data

and the one from Wagner and Pruÿ (2002), eq. 5.47. Furthermore, data is shown from Wagner and Pruÿ (2002) and data used in the 1Dradiative-convective model described in Sec. 5.2. One can see that all expressions and data compare reasonably well, except for a small deviation at very high temperatures for the formulation of Sonntag (1994). The inuence of this change is evaluated in Chapter 6 for the Earth around the Sun.

The replacement of this expression allowed for more stable model calculations, especially for the model scenarios Earth-like planets around the K-type star, which result in much higher surface temperatures. Nevertheless, it should be noted that for these scenarios, occasionally temperatures higher than 647 K occurred during the model spin-up due to numerical instabilities (even for small time steps of 120 s) occurred. In these cases the saturation water vapor pressure over liquid water for 647 K was used. As soon as the model atmosphere reached a new equilibrium state, such high temperatures were no longer obtained.

5.3.6 Orography parametrization

A sub-grid scale orography parametrization is implemented in EMAC to account for oro-graphic variations on scales smaller than the horizontal resolution of the model. The two main eects of orography on the atmospheric ow are the transfer of momentum from the ground to the atmosphere by gravity waves and the drag to the atmospheric ow which is caused by mountains which block the air ow at low atmospheric levels. The sub-grid orog-raphy is described by seven parameters linked to the orogorog-raphy, the standard deviation,

a critical level, when the ow does not lie in the plane of the low level stress, hence the gravity stress turns zero. The drag produced is the vertical derivative of the gravity wave stress. The breaking of gravity waves is parametrizedbasedon the Richardson Number and on the assumption that gravity waves reach saturation. When the height dependent Richardson number of the ow reaches the critical value of 0.25, gravity waves are assumed to be saturated. The gravity wave stress is then set constant to the saturation value for layers above until the saturation condition is met again.

5.3.7 Gravity wave spectrum

In addition to the gravity waves generated by orography described above, gravity waves can also be excitedby e.g. convective activity or front systems. Since the sources of gravity waves mostly reside in the troposphere they mostly only propagate through the middle at-mosphere. While dissipation of vertically moving gravity waves is taken into account by the model, horizontal movement of gravity waves on scales larger than the horizontal resolution is neglected. The deposition of momentum ux by vertically propagating gravity waves is parametrizedfollowing the Hines Doppler Spreadtheory. The depositedmomentum ux depends on the large scale ow, the buoyancy frequency, the input gravity wave spectrum andthe horizontal windvariance. Furthermore, parameters relatedto the gravity waves have to be calculatedby the parametrization such as the cuto vertical wavenumber, which species which wavenumbers of the input spectrum can continue to propagate upwards.

5.3.8 Surface processes 5.3.8.1 Surface albedo

The mean surface albedo of a grid cell depends on the fraction of land and sea surfaces (f_sea) in the gridcell andis basedon a backgroundalbedo (Abg) which has been derived from satellite observations of the Earth. The mean surface albedo A_surf is given by

A_surf =f_seaA_sea+ (1−f_sea)A_land (5.48) For sea surfaces the sea ice fraction (fseaice) is taken into account andthe albedo of the sea (A_sea) is calculatedfrom:

A_sea =f_seaiceA_snow+ (1−f_seaice)A_opensea, (5.49) whereA_openseais the albedo of sea water (set to 0.07) andf_seaice is either an input param-eter or calculatedby the ocean module (see section 5.3.8.6). A_snow is the albedo of snow

5.3. EMAC - 3D MODEL 81 or ice. It depends on the surface below the snow/ice cover. The albedo of the snow/ice covered ground is assumed to depend linearly on temperature, where lowest temperatures lead to lowest snow albedo and vice versa.

For land surfaces it is taken into account how much of the grid cell is covered with forest and/or snow.

A_land=f_{f orest}A_{f orest}+ (1−f_{f orest})A_ground (5.50) with

A_ground=f_snA_snow+ (1−f_sn)A_bg (5.51) and

A_{f orest}=fsvA_ground+ (1−fSV)Acanopy (5.52) wheref_{f orest}is the fraction of forest,A_{f orest}the albedo of the forest,A_ground the albedo of the forest free land surface, f_snthe snow cover of the ground. f_sv is the so-called sky-view factor, which is a measure of the visibility of the ground through a forest. The sky-view factor depends on the type of forest present, which is expressed in the leaf area index, which is and input parameter. The albedo of the canopyA_canopy is dened as

A_canopy =f_csnA_csn+ (1−f_csn)A_csnf (5.53) where f_csn is the snow load of the canopy which depends on temperature, wind and the type of forest (see also the paragraph about surface water reservoirs below). Acsn is set to 0.2. A_csnf is the albedo of the canopy without snow which is an input parameter.

5.3.8.2 Surface uxes

In the lowest model layer the surface uxes of momentum (J_u, J_v), static energy (J_s), moisture (Jq_v) and other water phases (Jq_w) are calculated, which depend on the drag coecient C_m, the heat transfer coecient C_h and the absolute value of the horizontal velocity |v_h|as follows:

J_u=ρC_m|v_h|u (5.54)

J_v =ρC_m|v_h|v (5.55)

J_s=ρC_h|v_h|(s−s_surf) (5.56) J_qw =ρC_h|v_h|(q_w−q_w,surf) (5.57) J_qv =ρC_h|v_h|(q_v−q_sat(T_surf, p_surf)) (5.58) for the moisture ux (evaporation) above sea and

Jq_v=ρC_h|v_h|((fsn+ (1−fsn)f_land)(qv−qsat)+

(1−f_sn)(1−f_land)(1−f_veg)(q_v−rhq_sat)+

(1−f_sn)(1−f_land)f_vegE(q_v−q_sat))

(5.59)

above land. Here, u and v are the horizontal velocity components, s the static energy, s_surf static energy at the surface q_w cloud water, q_w,surf cloud water mass mixing ratio at the surface, which is set to zero, q_v is the specic humidity and q_sat(T_surf, p_surf) the saturation water vapor specic humidity at the surface. For the moisture ux over land the fractions of land covered by snow (f_sn), land with a water reservoir (f_land), land covered with vegetation (f_veg) and land with bare soil are taken into account, since it is assumed

where C_L is the heat capacity of land in Jm⁻²K⁻¹, F_LH the latent heat ux, F_SH the sensible heat ux,F_GH the ground heat ux andF_RAD,net the net radiative ux which is:

F_RAD,net= (1−A_surf)F_s,↓+F_l,↓−σT_surf⁴ (5.61) with Asurf the surface albedo, F_s,↓ the downwelling shortwave stellar radiation, the surface emissivity Fl,↓ the downwelling longwave radiation and σT_surf⁴ the total energy ux due to the surface temperature via the Stefan Boltzmann's law.

5.3.8.4 Water reservoirs

The water reservoir of the surface is determined e.g. by the snow and rain at the surface and amount intercepted by the canopy. It is assumed that a quarter of the snow or rainfall is intercepted by the canopy. The amount of water stored then depends on the sublimation and deposition of snow or the evaporation or condensation (dew deposition) of liquid water, the unloading of the canopy by either slipping of snow due to wind speed or melting/evaporation of snow and rain depending on temperature. Melting of snow or evaporation of liquid water at the canopy cools the lowest surface layer depending on its pressure thickness. The amount of water which can be stored in the canopy depends on the leaf area index (LAI). Melting snow is converted into liquid water and then adds to liquid water reservoir of the canopy or to the surface reservoir. The remaining three quarters of precipitation reach the surface. Snow is either deposited, sublimated or melts, depending on the surface temperature (Eq. 5.60). Furthermore, the snow which slipped fromthe canopy adds to the snow reservoir. For the liquid water reservoir, the soil wetness, additional processes are taken into account such as the surface runo and drainage.

5.3.8.5 Lakes

Lakes in the model are assumed to be either frozen or ice free. Furthermore, a constant lake depth of 10 m(h_m) is assumed. The temperature of a lake is the determined by

Cw

∂Tw

∂t =H (5.62)

whereH is the net surface heat ux, including radiative and turbulent heat uxes,Cw = c_qρ_wh_m is the heat capacity of the lake, which depends on the heat capacity and density of water and the thickness of the lake.

5.3. EMAC - 3D MODEL 83