1. Progress in the implementation of the adjoint of the Ocean model NEMO by using the YAO software M. Berrada, C. Deltel, M. Crépon, F. Badran, S. Thiria Work supported by the SHOM (Hydrographic and Oceanographic Department of the French Navy) under SINOBAD project University of Paris VI LOCEAN Laboratory 9th European AD Workshop INRIA Sophia-Antipolis, 26-27, November 2009

2. NEMO (Nucleus for European Modeling of the Ocean) is a platform for numerical modeling of ocean The Ocean model NEMO NEMO Ocean Model Ocean Circulation OPA Model of sea ice LIM Marine Biogeochemistry LOBSTER, PISCES

3. Idealized configuration of the physical part of NEMO The domain is a limited area in the North of the Atlantic ocean (Gulf stream region) The horizontal dimension 32x22 and 31 vertical levels GYRE area localisation GYRE configuration

4. The ocean trajectory depends on the initial state Accurate initial ocean environment (V 0,ssh 0,T 0,S 0 ) V=(u,v) velocity ssh the sea surface height T Temperature S salinity Variational assimilation (YAO software) x=T 0 : Control vector y=M(x)=ssh at t= T : Observations (twin data) Cost function J(x)=||y-y obs || 2 A validation experiment with twin data

5. YAO Semi- automatic generator of the adjoint code Based on a modular graph structure The modular graph is a data flow diagram which describes the underlying physical model It consists of a set of modules, where the input of each one is provided by the output of its predecessors

6. YAO: Modular Graph M1 x11 y12 y11 x31 x32 x33 y32 y31 x21 x22 y21 Forward model d d d d d d d d d d d M3 M2 Backward model M’2 1. Defined the modular graph structure of the model 2. Coding of the local functions f q 3. Coding of the Jacobian F q Modular graph in a point of the grid

7. Modular graph of the space discretization M1 M2 M3 M1 M2 M3 M1 M2 M3 M1 M2 M3 M1 M2 M3 xx xx t0t0 YAO: Modular Graph

8. Accomplished work M1 x11 y11 y12 1. Defined the modular graph structure of the GYRE model under YAO 2. Coded the forward model 3. Implementation of the Jacobian of each module which is needed for the backpropagation (used for the computation of the cost function gradient ) d d d

9. Accomplished work Comparison: GYRE-YAO vs GYRE-Fortran (accuracy 10 -11 ) Comparison of the ssh at t=100 (1 time step =2h)

10. Linear tangent test α(k+1)=α(k)/2 1st order test 2nd order test : α (1): 2.500000e-02 : -1-> 7.914089e+00 -K-> 3.121720e+02 : α (2): 1.250000e-02 : -1-> 1.015254e+01 -K-> 8.051761e+02 : α (3): 6.250000e-03 :-1-> 3.269313e+00 -K-> 5.011360e+02 : α (4): 3.125000e-03 : -1-> 4.561872e+01 -K-> 1.460673e+04 : α (5): 1.562500e-03 : -1-> 8.877574e+01 -K-> 5.683726e+04 : α (6): 7.812500e-04 : -1-> 1.205476e+01 -K-> 1.542101e+04 : α (7): 3.906250e-04 : -1-> 1.799721e+02 -K-> 4.606209e+05 : α (8): 1.953125e-04 : -1-> 6.400455e+01 -K-> 3.278745e+05 : α (9): 9.765625e-05 : -1-> 3.628549e+03 -K-> 3.715594e+07 : α (10): 4.882813e-05 : -1-> 7.207983e+03 -K-> 1.476187e+08 : α (11): 2.441406e-05 : -1-> 9.999715e-01 -K-> 2.649493e+01 : α (12): 1.220703e-05 : -1-> 9.999856e-01 -K-> 2.674571e+01 : α (13): 6.103516e-06 : -1-> 9.999939e-01 -K-> 2.245280e+01 : α (14): 3.051758e-06 : -1-> 9.999962e-01 -K-> 2.876924e+01 : α (15): 1.525879e-06 : -1-> 9.999983e-01 -K-> 2.558008e+01 1st order test2nd order test

11. Adjoint test

12. Conclusion Flexibility: Modifying the model at any time is straightforward due to modular graph structure One can consider a more complex function as a module for the YAO graph and uses Tapenade (or other) to get the local adjoint Encouraging results of the adjoint tests

13. Thank you!