Figure 7.6: lbl validations: RRTM simple atmosphere (black) against SQuIRRL (red). Shown are uxes (upper panels) and ratios RRTM/SQuIRRL (lower panels).
Fluxes at 5 (left) and 50 (right) km altitude
However, given that the amount of radiation emitted in this band is very low, compared to the overall ux, its inuence on the stratospheric cooling rate is negligible and was found to be less than 10−3 for all model scenarios. Hence, the error in the calculated temperature prole is correspondingly small.
Table 7.2 summarizes again the results of the lbl validations of MRAC and the runs performed.
Table 7.2: lbl-validations: deviationsσlbl of models to lbl results
Scenario Totalσlbl [%] No of bands withσlbl>10% Maximumσlbl [%]
RRTM modern 5 km 2.2 3 28
RRTM simple 5 km 2.1 1 15
MRAC nocont 5 km 0.2 4 17
MRAC allcont 5 km 0.2 4 19
RRTM modern 50 km 0.7 1 17
RRTM simple 50 km 1.1 3 45
MRAC nocont 50 km 3.5 8 76
MRAC allcont 50 km 2.8 6 68
7.3 Boundary, initial conditions and numerical scheme in
7.3.1 Inuence of boundary conditions
The boundary condition is determined by (i) the pressure of the upper model lid, p0 and (ii) the downwelling uxes at this upper model boundary, F|p=p0. In the model, the value of p0 is usually chosen as p0= 6.6·10−5 bar (mid-mesosphere). The downward components of solar (stellar) and thermal ux are set by the user (see Table 6.8). Fuser,sol is given by the assumed solar constant and the type of central star, and Fuser,therm is set to 0 Wm−2. This latter condition means that from abovep0, no atmospheric thermal radiation is emitted downwards.
Figure 7.7: Temperature proles for dierent pressure boundary conditions: Con-trol (p0=6.6·10−5bar, black),p0=10−3 bar (red),p0=10−4bar (green), p0=10−5 bar (blue) andp0=10−6 bar (magenta)
Fig. 7.7 shows the results of simulations with changed boundary conditions.
The value ofp0 was set to 10−6, 10−5, 10−4 and 10−3 bar, respectively. Lower atmospheric conditions and surface temperatures did not change signicantly, indicating that the choice of the boundary condition has little inuence on the lower atmosphere, as expected. Of course, upper stratospheric conditions changed quite dramatically, depending on the choice of p0. Still, the overall shape of the temperature prole remained unchanged.
7.3.2 Inuence of initial temperature conditions
Usually, the initial temperature prole is the standard modern Earth prole based on the US Standard Atmosphere 1976 (see, e.g., Segura et al. 2003). To test how much the choice of initial temperature prole aects the result, it was changed to allow for an isothermal start (i.e., all layers started with the same temperatureTI). Runs were performed for several values ofTI (200, 250, 288, 300, 400, 500 and 600 K). No signicant deviations of calculated temperature
proles were found, indicating that the choice of the initial temperature prole is not important for the nal solution.
7.3.3 Inuence of time step
The termdt in eq. 6.3 should not be interpreted as a real, physical time step.
Even though it has the dimension of [seconds], it is however not an evolutionary time step, as for example in 3D models. In this 1D model, the perturbed system is a globally and diurnally averaged system, hence no physical time is considered in the solution of the numerical equations. The time step adjusts itself according to calculated heating and cooling rates and the temperature changes. If changes between consecutive iterations are relatively large, dt is decreased, however if small temperature changes indicate that the system is close to a solution, the time step is increased. Thus, the value of the time step is always adjusted to the particular system.
As described in section 6.3, the temperature prole is smoothed in the strato-sphere (see eq. 6.8). If the time step is too small, then this smoothing dom-inates the calculation of the temperature prole. Physical eects such as ra-diative transfer or convection introduce temperature changes which are small compared to the smoothing. If, however, the forced time step dt is too large, temperature uctuations between consecutive iterations become too large and the model fails to produce converged or stable solutions.
To illustrate this and test the inuence ofdt on the nal results, the maximum dt value in standard model runs (usually reached within a few iterations) was articially forced to values between 102 and 106 seconds. The default value is 105 seconds. Results are shown in Fig. 7.8.
Figure 7.8: Temperature proles for dierent time steps in the model. Value of dt in s as indicated, for the control rundt=105 s
It demonstrates that the choice of the maximal dthas indeed a large inuence
on the resulting temperature prole, as expected.
For the reference conditions used here, the time step should be of the order of 105 seconds. This can be inferred from Fig. 7.8. For small values of dt (102 and 103s), no troposphere develops. When dt is increased to 104 s, a troposphere with convection is calculated, however stratospheric conditions do not yet react to the radiative transfer. Only at values of dt of 105-106 s, the stratospheric temperature prole is inuenced by radiative processes.
7.3.4 Inuence of vertical grid
The vertical grid in the model is controlled by two parameters, the number of levels (parameter ND) and the grid spacing ratio between top and bottom of the model atmosphere (parameter FAC). In the current version of the model, the values are ND=52 and FAC=2.5 which means that the grid spacing at the bottom is 2.5 times ner as in the upper atmosphere. The inuence of these parameters on the model results are investigated.
Vertical grid spacing
Fig. 7.9 shows the results for variations of FAC, with FAC=0.5 up to FAC=5.0, compared to the control run with FAC=2.5.
Figure 7.9: Temperature proles for dierent values of parameter FAC, as indicated.
Control: FAC=2.5
The eect on surface temperature is small, with an increase of about 0.8 K when increasing FAC from 0.5 to 5. This was related to the fact that with smaller values of FAC, less points are available in the lower atmosphere, hence the location of the tropopause was less accurately determined. With increas-ing value of FAC, the surface temperatures increased as the location of the
tropopause moves slightly upwards, hence more surface warming due to con-vection was produced.
Number of vertical grid levels
The same eect can be observed when increasing the number of vertical levels, with the value of FAC unchanged. This is shown in Fig. 7.10, where the value of ND has been increased from 13 to 156.
Figure 7.10: Temperature proles for dierent values of parameter ND, as indi-cated. Control: ND=52
The eect on surface temperature is relatively small, as was the case for the FAC changes. Temperatures increased with increasing ND for the same rea-son as they did with increasing FAC. The location of the tropopause could be resolved with increasing accuracy. However, the largest changes could be observed in the stratosphere. With increasing number of grid points, the tem-perature prole seems to converge towards one value which diers quite signif-icantly from the one of the control run (black line in Fig. 7.10). The change between ND=156 and ND=208 is relatively small. However, runs with a still higher number of grid points (e.g., ND=312), are not shown since the model did not obtain a converged solution as dened by the conditions in section 6.3.
This is due to two reasons.
Firstly, the boundary condition assumption (i.e.,F|p=p0=0 Wm−2 for the IR) becomes increasingly invalid at higher vertical resolution, hence the conver-gence problems.
Secondly, the temperature in the bottom layer of the atmosphere is calculated via (see eq. 6.4):
d
dtTb =− g cp(T,bottom)
dFTOA
dpbottom (7.1)
Ifdpbottom becomes small (with increasing vertical resolution,it decreases from
∼0.2 bar to∼0.01 bar), the calculated temperature in the bottom atmospheric layer becomes unrealistically large. This equation is not suited to simulate very thin surface layers due to the term dFTOA/dpbottom.
This implies that there is a maximum vertical resolution which the model can use in order to calculate converged temperature proles.
Rened upper boundary condition
The maximum vertical resolution found in the previous section depends on the assumed CO2 concentration in the model atmospheres. This is due to the choice of the boundary condition F|p=p0=0. This introduces articial cooling rates because above the model lid, generally F|p>p0 ̸=0.
For present-day Earth concentrations, this is not signicant since uxes are already very low. However, for higher CO2 concentrations this articial cool-ing rate can become the dominant contribution to the radiative budget. For CO2 concentrations of about a few percent, a doubling of vertical levels (i.e.
ND=104) still allows for convergence in the stratosphere, whereas for higher CO2 concentrations (∼ 50-100 %), even ND=52 is too large to achieve formal convergence of the stratospheric temperature prole.
The overall temperature proles calculated for such scenarios do however pro-vide meaningful results for the lower atmospheric and surface conditions. This is demonstrated in Fig. 7.11 for an example case of a 10 bar CO2 atmosphere with additional 800 mbar of N2, at a reduced solar constant of S=0.8. Result-ing proles for two dierent numbers of iterations (1,000 and 2,000 iterations) are shown. These proles dier in the upper atmosphere, therefore the proles are not converged, as stated above. However, as is also clearly seen, in the lower atmosphere, and especially at the surface, temperature proles do not show signicant variations. This implies that the atmospheric structure at pressures above about 10−4 bar is stable even for high CO2 values.
Figure 7.11: Stability of temperature proles: Run with 1,000 iterations (plain line) and 2,000 iterations (dotted line).
In order to investigate further this convergence problem, the implementation
of the upper boundary condition was changed. As stated above, it isF|p=p0=0 Wm−2for the IR, hence, downwelling uxes are set to 0 at the upper boundary.
In reality, this is only true for optically thin layers. If the optical depth is large (i.e., τ ≫ 1), the atmosphere layer will emit radiation according to the local temperature. Therefore, numerical tests were performed where the upper boundary condition was set to I|p=p0=B(Tlocal) (B(T) the Planck function) if τ >2.3, which is equivalent to a layer transmission lower than 10 %.
However, tests with this rened boundary condition did not result in an im-provement of the convergence behavior. This is partly due to the fact that this type of boundary condition involves the local temperature, i.e. the quantity to be calculated with eq. 6.3. Hence, the boundary condition is no longer constant during the calculations, which poses a principal numerical problem.
Given that atmospheric proles are still stable (see Fig. 7.11), this is a minor problem for the present model.
Based on the above investigations, the new reference model still uses the orig-inal vertical grid resolution ND=52.