Verification of time integration for non-volatile species¶

Time convergence test of the calculation of sulfuric acid gas and sulfate aerosol.

Test code can be found in the mam_refactor code repository under standalone/tests/gasaerexch/.

Test design¶

Tests are set up to verify the convergence of sulfuric acid gas and sulfate aerosol in MAM for the following aspect:

In MAM, the analytical solution is used to solve the governing equation of sulfuric acid gas condensation, which can be machine precision with constant uptake rate.
With updated uptake rate for each timestep, first order convergence is expected for the linearized equations.

Test driver¶

The test driver file is skywalker_gasaerexch_timestep.F90 and skywalker_gasaerexch_timestep_update.F90 in the mam_refactor code repository, located under standalone/tests/gasaerexch/

The time integration of sulfuric acid gas and sulfate aerosol is implemented under core/modal_aero_gasaerexch.F90 mam_gasaerexch_1subarea_1gas_nonvolatile subroutine. Because the non-volatile species and semi-volatile species are calculated seperatedly in the mam_gasaerexch_1subarea subroutine and uptake rate need to be derived before the time integration, tests are conducted on the mam_gasaerexch_1subarea subroutine.

Two sets of tests are conducted to verify the time convergence of sulfuric acid gas and sulfate aerosol.

In skywalker_gasaerexch_timestep.F90 test driver, we test on case where uptake rate remains constant for each timestep.
In skywalker_gasaerexch_timestep_update.F90 test driver, we test on case where dry diameter of aerosol is updated according to the current aerosol concentration, namely uptake rate is updated every timestep as the uptake rate calculation depends on aerosol size.

Input parameters¶

Input parameters are specified using YAML files stored in the mam_x_validation repository. The YAML files are imported to each clone of the mam_refactor repository as part of a submodule. The YAML files for the tests described on this page are located under standalone/tests/mam_x_validation/gasaerexch/gasaerexch_timeconvergence.yaml and standalone/tests/mam_x_validation/gasaerexch/gasaerexch_timeconvergence_update.yaml
Different pressure, temperature and timesteps are used to generate input ensembles. Details are below:

Input ensemble

Variable	Range
Temperature(K)	180, 200, 233, 273, 298
Pressure(Pa)	1000, 100000
Timestep(s)	0.1, 1, 2, 4, 10, 50, 100, 300, 900, 1800

Because the $H_{2}SO_{4}$ will deplete within 30min simulation time. A production rate of 10 ppb/timestep $H_{2}SO_{4}$ gas is assigned in the test to ensure the consistent production of sulfate aerosols from H_{2}SO_{4}$ condensation. Time series plot of $H_{2}SO_{4}$ and sulfate aerosol without(Figure S1) and with(Figure S2) $H_{2}SO_{4}$ gas production are shown here.

Test output¶

After executing the verification tests, the output data and plots can be found in your build directory under standalone/tests/gasaerexch/.

The plots are generated using the script standalone/tests/mam_x_validation/gasaerexch/plot_gasaerexch_timestep.py and standalone/tests/mam_x_validation/gasaerexch/plot_gasaerexch_timestep_update.py.

Reference results¶

Figure 1 shows the result of time convergence test for $H_{2}SO_{4}$ and sulfate aerosol. X-axis denotes the timestep and y-axis is the relative error. Both $H_{2}SO_{4}$ and sulfate aerosol show machine precision at varied timestep, which matches the feature of analytical solution to the linearized equation with constant coefficient (uptake rate).

fig — Figure 1. Relative error of the time convergence test for H₂SO₄ and sulfate aerosol.

Figure 2 shows the time scale of $H_{2}SO_{4}$ and sulfate aerosol condensation. E-folding time of uptake process is short (on the order of seconds) at different temperature and pressure levels, except for the coarse mode.

fig — Figure 2. E-folding time of uptake rate.

Figure 3 shows the test results with dynamic dry diameter update at each timestep. The regression slopes are 1.05 and 1.05 for $H_{2}SO_{4}$ and sulfate aerosol correspondingly at dt = 1s, 2s, 4s and 10s.

Future work

It is worth revisiting MAM’s time integration method for condensation with dynamic wet diameter update at each timestep.

fig — Figure 3. Relative error of the time convergence test for H₂SO₄ and sulfate aerosol with updated uptake rate.

Supplement Figures¶

fig — Figure S1. Time series of H₂SO₄ and sulfate aerosol without H₂SO₄ production.

fig — Figure S2. Time series of H₂SO₄ and sulfate aerosol with H₂SO₄ production.