1. Henry Juang1 Non-iteration Semi-Lagrangian: mass conservation Hann-Ming Henry Juang Environmental Modeling Center, NCEP NOAA/NWS April 18, 2007

2. Henry Juang2 Introduction The new semi-Lagrangian scheme –Without iteration to find departure/mid points –Combine two 1-D interpolation/remapping –Without halo in global model –Encouraging results from regional/global tests The concerns –Can we have conservation? –Positive advection?

3. Henry Juang3 For mass conservation, let’s start from continuity equation Consider 1-D and rewrite it in advection form, we have Advection form is for semi-Lagrangian, but it is not conserved if divergence is treated as force at mid-point, So divergence term should be treated with advection

4. Henry Juang4 Divergence term in Lagrangian sense is the change of the volume if mass is conserved, so we can write divergence form as Put it into the previous continuity equation, we have which can be seen as

5. Henry Juang5 ADM Interpolation remapping relocation Instead doing following

6. Henry Juang6 ALAL DLDL M Interpolation relocation We do X DRDR ARAR MLML MRMR

7. Henry Juang7 Since we are using cubic spline, the given value can be presented piece-wisely by so the previous mass equality can be replaced as following Also we want to make sure that total mass is conserved as This implies that mass conservation should be used during interpolation from regular grid to departure grid and from arrival grid to regular grid. where subscript R is regular grid D is departure grid A is arrival grid for

8. Henry Juang8 Interpolation with global conservation XXXX XXXX

9. Henry Juang9 Summary of the procedure (1) Using and cubic spline conditions, we can solve (2) Using semi-Lagrangian mass conserved advection as following and cubic spline conditions at arrival points, we can solve (3) Using mass conservation interpolation as following and cubic spline conditions at regular point, we can solve (4) With we can have

10. Henry Juang10 Cubic Spline Interpolation/Remapping Definition of piece wise cubic spline curve fitting with periodic condition which requires 4 equations to solve all 4 unknown coefficients. In case of interpolation, such as the 4 equations are In case of remapping, such as the 4 equations are the same as above, except replace the first one by or where

11. Henry Juang11 How about mass conservation for tracer ? If we use tracer and continuity equation as following together Then density weighted tracer can be treated as conservation as Combine it with continuity equation, we can have conserved tracer advection

12. Henry Juang12 Summary Mass conservation can be obtained –No guess, no iteration to find departure/mid points –No halo is required but transpose, which exists in the code already –Cheap1-D interpolation/remapping –Local and/or global conserved Test simple cases with success – tracer advection and steady rotational flow