The investigations of the behaviour of the gradients at high resolution have been continued on the one hand to distinguish which was the mechanism for developing strong punctual sensitivities over the Alps in the so-called 'The Second French Storm' or 'T2' storm, and on the other hand the using of linearized simplified physical parameterization in adjoint ALADIN model. The accuracy of the linearization of the simplified physical parameterization has been compared with the nonlinear results.
When using a model integration area with high resolution, one of the characteristics is a better described orography. Also the parameterization of physical processes has to be able to simulate mesoscale feature of the atmospheric flow. During a strong advection, if high mountains barrier is encountered, strong gravity waves are generated. In this case, a numerical model will produce even higher energy coming from gravity waves than in reality. With a proper time step and a good physical description (e.g. surface drag and vertical diffusion) this energy is dissipated and the model remains stable.
For avoiding numerical instability in the computation of the gradients, the adjoint of a direct model needs also a proper physical description of the atmospherical processes. The difficulty of developing a complex adjoint model arise not from the linearization of the primitive equations but from the physical processes due to the strong nonlinearities and discontinuities. Buizza (1993) showed that an important physical parameterization for the adjoint models is the planetary boundary layer scheme. In our experiments, using adjoint of the ALADIN model, the results have shown that for a coarser domain (e.g. Irish Christmas Storm), the simple representation of the vertical diffusion and gravity wave drag was enough to prevent the development of non-physical structures in PBL. When using a high resolution area, this poor linear physics is no more sufficient.
The results for T2 storm have revealed very strong gradients without meteorological relevance. To get rid of this spurious structure, several sensitivity studies were followed, we recall the reducing of the Eulerian integration time-step from 40 to 10 s having no effect and, the changing of the trajectory truncation in the adjoint model. Also, a retuning of boundary layer height and the scale factor for the wind speed in the crudest linearized simplified physics has been performed. The results, has revealed that the later approach is more appropriate but the effect was not only on the non-meteorological structures but also on useful part of the gradients.
Trying to answer the question why they have appeared we have searched for a key in the 18-hours forecast. A temporal sequence of spatial vertical cross-section along a relative direction where the gradients with the biggest magnitude have occurred have shown at +6h the existence of a strong baroclinic region in the upwind part before the mountains. Immediately downwind there is a turbulent region as suggested the form of the isentrops. Using the adjoint model from +18 hours to +0, the gradients of the forecast error norm were computed. As the normal flow is westward, in adjoint the wind tend to advect perturbations eastward. The results has demonstrated that with the 60 s time-step, the model becomes unstable in the last few hours of backward integration, the magnitude of the gradients having an exponential increase. An explanation is that the strong wind existing in the last hours of the integration corroborated with the improper time-step have produced a model stability criterion violation. Since for 40 s the instability has vanished, although big but not unrealistic sensitivities have developed, it appeared the necessity of a better description of the physical processes in the adjoint model.
The more complex simplified physical parameterization scheme proposed by Janiskova et al. is based on the vertical diffusion equation with a first-order closure for the exchange coefficients. During their tests some spurious noise close to the surface have been noticed. It was discovered that this problem was derived from the vertical diffusion scheme, and especially from the function of Richardson number, f(Ri). To overcome the occurrence of the artificial noise, a modification of f(Ri) to ensure a smaller derivative by a factor 10 in the central part, around the point of singularity, Janiskova et al. tested and adopted for the tangent linear version of the vertical diffusion scheme.
Including the vertical diffusion and gravity wave drag schemes in the ALADIN adjoint model, the results after 3h of integration have shown a proper magnitude and pattern for the computed mesoscale gradients. Using a completely adiabatic adjoint model, the computed sensitivities have big values mainly under the model level 20, about 2800 m. This result also suggests the necessity of a better description of PBL processes in adjoint model. When the simple linearized physics is included with default values for the boundary layer height and the scaling factor for the wind stress, the gradients values diminish compared to the adiabatic case, but remain big enough not to be used like small perturbation in the initial conditions. Yet, the simplified physics produces the smoothest vertical profiles and they are the most efficient in damping the undesired spots of gradients over mountainous area. With retuned values, the simple linear physics produce vertical profiles close to those of the simplified physical parameterization case. Unfortunately, it was not possibly to carry out the 18h adjoint integration using simplified physical parameterization since it requires a huge computer memory for storing of the trajectory. Nevertheless the aim of this linearized physics is not so much for 18h sensitivity studies but for no more than 6h mesoscale gradients computation which should be useful for a future 4d-var assimilation in ALADIN model.
In fact, even if a posteriori, the improvement of the precipitation forecast could be seen as one of the latest but not the last challenge of using the mesoscale sensitivities. The questions to be answered include: (1) Could be improved a bad forecasted precipitation field without modifying the model initial humidity state variable ? (2) Using the gradients method for changing the initial conditions, it could be possible to re-forecast precipitation where the model was not able to produce but they did occur in reality? Maybe one of the questions should have the answer expected from simple consideration of model dynamics or physical parameterization. However, we are not going to search the answer in this part, but primarily in the analysis errors.
As a case study, two spring situations have been used. We consider they are relevant for our purpose. They were selected because the precipitation forecast was bad, both amount and localisation. The integration domain used in the following experiments was ALADIN-France operational, (300´300´31, Dx=9.5 km). The gradients of 6 hours forecast errors have been computed starting from the verification time, 06 UTC. By adjusting the initial model state with a fraction of the gradients, the final purpose was the improvement of the 6-hours precipitation forecast.
Using the Taylor formula the global mathematical correctness of the linearized code of the simplified physical parameterization for ALADIN model with respect to the nonlinear version was done. Starting from a defined initial state, x, the accuracy of the Taylor formula is computed in the vicinity of a trajectory which is a nonlinear forecast. The testing perturbation, dx, is chosen at random. The standard validation of the tangent linear code have been performed for 7 steps, with 60 s time-step. The gravity wave drag, vertical diffusion and stratiform precipitation schemes were successively switched on. The results have shown a good accuracy of the linearization when using the schemes individually or in combination.
Comparisons have been done for the 3-hours evolution, on one hand of perturbation computed with the tangent linear model, and on the other hand of the finite differences between two nonlinear predictions using the simplified physical parameterization package. If taken individually, the linearization is valid for each scheme. Also when combining vertical diffusion with gravity wave drag scheme.
The results show that the nonlinear model become unstable for the small perturbations. The degree of the instability of nonlinear model is case dependent. For a good linearity when using large scale precipitation scheme together with vertical diffusion and gravity wave drag, one has to activate the smoothing functions.
The two synoptic cases considered in our study have a general feature namely the 6-hours time window for gradients computation. This was the limit enforced on one hand the technical reasons and on the other hand the requirement to work within a time window which is supposed to be used in a future variational assimilation.
For the experiments using the adjoint model, we were concerned with sensitivity pattern when including simplified physical parameterization for gradients computation. The results for several tests have revealed that a parameter having a strong influence in gradients computation is the minimum critical thickness for precipitating clouds (ECMNP inside the code). Moreover, as long as convective scheme is not included in the adjoint model, it contributes to the trajectory only. This makes the adjoint model including large scale precipitation scheme to be much more sensitive to the physics used for creating trajectory than an adjoint with the dissipative part of the physical parameterization.
Several experiments using simplified physical parameterization as well as tangent linear and adjoint versions have been carried out. Due to technical reasons, the time window chosen in our tests was mainly 6 hours.
A first set of experiments performed for the Second French storm, have revealed that the spurious gradients structure developed over the Alps when the simple physics is included in adjoint model, vanish if simplified physical parameterization scheme (vertical diffusion and gravity wave drag) is used. This result obtained for a 3-hours time window has demonstrated the necessity of including a more sophisticated physical parameterization in the adjoint model.
The validation of the linearization of the simplified physical parameterization, that is the dissipative part plus large scale precipitation, has been performed as well. The results have shown that the tangent linear model describing the evolution of a perturbation of the order of magnitude of analyse increments fit the finite differences between two nonlinear forecasts. The nonlinear model become unstable at small perturbation. To obtain a good accuracy it is necessary to use the smoothing function both in direct and tangent linear models.
An adjoint model including a more sophisticated description of the physical processes become very sensitive to the trajectory. This is the result obtained when in adjoint model, large scale precipitation scheme was included besides vertical diffusion and gravity wave drag. Also it has been revealed that even a posteriori it is very difficult to ameliorate a failure of a precipitation forecast without a direct changing of the specific humidity state variable. Future work will concentrate in the direction of improving the model humidity field using the adjoint method.
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