© Crown copyright Met Office Adaptive mesh method in the Met Office variational data assimilation system Chiara Piccolo and Mike Cullen Adaptive Multiscale Methods for the Atmosphere and Ocean 23 August 2012

© Crown copyright Met Office Contents This presentation covers the following areas Adaptive grid in 3D-Var: formulation Adaptive grid in 3D-Var: application Initial implementation and limitations Improved implementation Summary and future work

© Crown copyright Met Office Adaptive grid in 3D-Var: formulation

© Crown copyright Met Office Adaptive Mesh Transform in 3D-Var Motivation: improving accuracy in areas of large spatial variations of the 3D-Var solution, e.g. strong temperature inversions due to the presence of stratocumulus clouds. Static Adaptive Mesh Methods relocate grid points in a mesh so that they concentrate the grid points in areas where there is a rapid variation of the atmospheric field. For this kind of method, interpolation is required to pass the analysis increments on the old mesh to the new mesh. The transformation from the physical grid to the computational grid is guided by a monitor function, which controls the mesh distributions. The choice of the monitor function is problem- and user-dependent and can be determined a priori from consideration of the geometry or the physics of the atmospheric feature we want to diagnose.

© Crown copyright Met Office 3D-Var Transforms  The adaptive mesh transformation aims at changing the vertical background-error correlations by moving the vertical levels to concentrate mesh points around temperature inversions.  The movement of the levels is guided by a scalar monitor function which here we choose to be a function of the static stability which strongly controls the vertical mixing of the atmosphere and thus probably the vertical correlation structure of model variables.  We introduce the adaptive method within the Met Office Var system as an another transformation in the sequence of variable transformations aimed to simplify the background term of the cost function: where U a is the "adaptive mesh transform" which is placed between the parameter transform U p and the vertical transform U v

© Crown copyright Met Office The first step of the U a transform is to calculate a monitor function M ( >0 ) in physical space z [0,1]: The second step is to generate the adaptive mesh in physical space by defining a computational coordinate  : Adaptive Grid Formulation The map from computational domain to physical domain is thus defined by the a unique one-dimensional map which connects intervals of a prescribed length. Finally, the control variables  which will be generated at points  by the vertical transform are then interpolated to the true levels z.

© Crown copyright Met Office Choice of the Monitor Function Lorenc (2007) showed that a significant part of the observed background-error correlation structure could be explained by regarding it as a function of static stability ( ). Most choices of monitor function need regularisation to perform effectively and ensure that a good mesh resolution is maintained everywhere, an example is given by: M is always positive and can be modulated by a scaling factor c. If the scaling factor c is set to zero, the computational grid and the physical grid are the same. Since mesh points will be clustered where the monitor function is large, this choice of M will cluster mesh points in regions of large static stability.

© Crown copyright Met Office Horizontal Smoothing The adaptive mesh transform is a 1D transformation in the vertical only. The transformation depends on horizontal position. The monitor function is calculated for every horizontal grid point. In order to avoid a loss of horizontal coherence in the mesh a horizontally smoother mesh can be generated by smoothing the regularised monitor function prior to the mesh calculation. The smoothing at point i, j (longitude/latitude) can be expressed as: The degree of smoothing applied can be increased by iterating this smoothing procedure N times.

© Crown copyright Met Office Adaptive grid in 3D-Var: application

© Crown copyright Met Office Monitor Function and Adaptive Grid

© Crown copyright Met Office Single Ob tests: Stratocumulus Nominal physical mesh Computational mesh Single Ob q T inserted above the inversion Computational mesh Nominal physical mesh Single Ob  inserted above the inversion

© Crown copyright Met Office Initial implementation: limitations

© Crown copyright Met Office RMS error: Analysis - Observations thetazonal wind relative humiditymeridional wind

© Crown copyright Met Office RMS error: surface observations RMST (K)RH (%)u (m/ s)v (m/s) Control Test N obs Improvement of analysis RMS error for temperature at the surface. Reduction of the background vertical correlation distance for the potential temperature and an increase of its variance Variance increase induced by increase of pressure vertical gradient resulting from the reduced vertical correlation distance. Near the surface less weight is given to the background state and more weight is given to the observations. Adaptive grid at Camborne

© Crown copyright Met Office Forecast RMS error for T 2m Results from the full coupled analysis/forecast system:  positive impact up to T+15 h for 2m temperature forecasts  relatively neutral impact for the other variables. The forecast error is reduced most at the beginning of the forecast. This reflects the movement of the grid and the local interaction between atmospheric variables near the surface. The impact greatly decreases as the lead time increases. CasePeriodUK index Winter8/2/ /2/2010Overall: +0.44% T 2m : +0.23% Spring25/4/ /5/2010Overall -0.08% T 2m : +0.07% Time series of the forecast RMS error verified against surface observations for the winter trial

© Crown copyright Met Office Background Adaptive Grid Standard Grid Observations Limitations  The monitor function is based on the background-state static stability.  The monitor function is calculated every analysis cycle and depends uniquely on the previous cycle forecast. This limits its effectiveness because it relies on the presence and position of low clouds or temperature inversions in the background-state.

© Crown copyright Met Office Improved implementation

© Crown copyright Met Office Updates to Adaptive Grid transform in 3D-Var  More appropriate normalisation factor  Recalculation of the computational grid based on the updated background-state in 3 steps: 1.standard assimilation for 10 iterations in order to get closer to the observations and improve the background-state (called AG0) 2.calculation of the monitor function from the preliminary analysis coming from step 1 and application of the adaptive grid transform in the assimilation for 10 iterations only (AG1) 3.second calculation of the monitor function and grid based on the improved analysis coming from step 2 and application of the adaptive grid transform in the assimilation until convergence (AG2)

© Crown copyright Met Office UK4: 00Z Feb 8 th 2010 Note: large horizontal variations of the monitor function!

© Crown copyright Met Office UK4: 3 Jan z MSG image at 3.9  m over UK for 3 Jan 2011 at midnight: emitted radiation from the surface and cloud tops (3km) White areas: cold temperatures, i.e. clouds Dark areas: warm temperatures, i.e. sea and land surfaces. High clouds over North Atlantic Ocean and west of Ireland Patches of clear sky over the sea and France Low clouds over all UK!

© Crown copyright Met Office Normalisation factor Control case Test case UK4 domain: 3 Jan z

© Crown copyright Met Office Recalculation of the grid LS = background-state (3h forecast) AG0: standard 3D-Var (10 iterations) AG1: LS1=LS+PF-AG0 (10 iterations) AG2: LS2=LS+PF-AG1 (convergence) LSLS1 LS2 UK4 domain: 3 Jan z

© Crown copyright Met Office Var analysis cycle from the full coupled analysis/forecast system: improvement of the background-state Single Var analysis cycle: monitor function calculation started from the same background-state Control Theta AG2 Control AG2 AG2 cycled RMS error: Analysis - Observations  [K] RMS Error UK4 domain: 3 Jan z

© Crown copyright Met Office PeriodVisPrecipCloud amount Cloud base TempWindOverall 23 Dec 2010 – 3 Jan %5.48%-1.05%3.03%0.22%-0.04%+0.25% 10 Aug Aug %0.00% 4.17%0.23%0.10%+0.55% All options tested provide a small positive impact to the UK index in winter. After extensive tuning, the best option in the full analysis/forecast system was given by using static stability with the recalculation of the grid: consistent signal for surface temperature and cloud base height. Analysis/forecast system performance when recalculating the grid twice These results are reinforced by a general improvement of the initial fit to the observations in all analysis cycles of few % for both seasons. Which observation types contribute the most?

© Crown copyright Met Office RMS error vs sondes: mean relative difference [%] to control case O-BO-A

© Crown copyright Met Office RMS error vs other obs types: mean relative difference [%] to control case black = O – B; white = O – A

© Crown copyright Met Office Winter: surface temperature Results from the full coupled analysis/forecast system:  improvement of RMS error for both winter and summer cases at the beginning of the trial and neutral afterward  improvement of bias error for winter only at the beginning of the trial and neutral afterward  relatively neutral impact for the other variables Analysis vs Surface Observations

© Crown copyright Met Office Analysis vs Sonde Observations Temperature RMS error over the winter period Results from the full coupled analysis/forecast system:  small improvement of temperature RMS error versus sonde profiles in the lower atmosphere for both winter and summer cases up to T+ 6h  relatively neutral impact for the other variables

© Crown copyright Met Office Background Analysis Observations 3 Jan z: Camborne The monitor function is based on the background state: if the inversion is not present, the vertical grid does not change. When the monitor function is based on an update background-state using the observation’s information in the minimisation process the analysis has a clearer inversion. Limitation: long horizontal and mode independent length scales in the background error covariance matrix. QC: RH sonde not assimilated around the inversion because observations are too far from background.

© Crown copyright Met Office Summary and future work

© Crown copyright Met Office Summary and future work Initial operational implementation of the vertical adaptive grid transformation in 3D-Var in November 2010 had an impact mainly in 2m temperature. In July 2011 the recalculation of the adaptive grid within the minimisation process became operational. The main limitation is due to the long horizontal length scales which are fixed for each vertical mode in the UK models. This washes out the local effects of the adaptive grid transform. More realistic horizontal length scales will allow to detect more local phenomena. In the future we will extend the transformation to full 3D:  see C. Budd’s talk tomorrow on: Monge ampere based moving mesh methods with applications to numerical weather prediction

© Crown copyright Met Office Thank you for your attention. Any questions?