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** Tangent Linear Validation **

## Principle

The tangent validation test checks that the Tangent Linear model **L** is a first order approximation of the model **M**. We can then write a second order Taylor expansion as follows:

M(X0+p.dX) = M(X0) + L(p.dX) + e

with,

X0: initial state vector dX: perturbation state vector p: scale factor e: residu

when the factor **p** tends to zero we have:

-1- M(X0+p.dX) - M(X0) tends to L(p.dX) and -2- e is equivalent to O(p^2), means Norm(e,.) <= K.p^2 (with K a constant)

This means that for a given **p0**, for p smaller than p0 the error of the Linear Tangent with respect to the model is decreasing as p^{2.
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### First order validation

Let's note

Np = M(X0+p.dX) - M(X0) Lp = L(p.dX)

the first order validation checks:

for a given p0, for all p>p0: Ep = Norm(Np,.)/Norm(Lp,.) tends to 1

### Second order validation

the second order validation ensure that the residu is controlled. We check then:

for a given p0, for all p>p0: Rp = Norm(Np - Lp,.) behaves as p^2

note that we can also perform the following equivalent tests:

test | expression | behaviour |

b | Norm((Np - Lp)/Lp,.) | p |

c | Norm((Ep - 1,.)/p | constant |

d | Norm((Ep - Lp/(p.Lp),.) | constant |

Note that the second order test is not relevant if **M** is linear. **L** must be exact and this test will only give information on the actual machine error.

## Implementation

We need to compute Np, Lp, Ep and Rp for different value of p. It means that we need to run several time the model **M**. As we wish not to modify the model, multiple run of the model currently means launched several times the executable. A dedicate driver is developp to be able to run in the same executable the model **M** and its tangent **L**. This driver is used as soon as the the logicals **ln_tst_tan** (tangent test switch) and **ln_tst** (general test switch) are set to .true. in the **namtst** namelist. (we also have to choose between testing the main routine **step** or individual options, as for adjoint test, with **ln_tst_tan_cpd** (swith for testing individual options) set to .false. or .true.)

### Process

The test is built as follows:

Loop on p: * IF p=0: * Write to file M(X0) * ELSE: * compute p.dX * compute M(X0+p.dX), L(pdX) * read M(X0) * write to file Np, Lp, Ep and Rp End Loop Concatenate outputs files

### How to Run the test

**runnemo_tgt.ksh** (only ORCA2_Z31 cfg) is a dedicated script to run tangent test of TAM.
Basics options are:

- t: defined working directory target
- A: compiler setting base
- j: direct trajectory file directory (default value is corresponding outer loop directory)
- v: experiment target (= nemotam)
- k: non-perturbated outputs directory
- f: diagnostic files directory
**note**: for 'k' and 'f' options, you must indicate the relative path with respect to 't' option

Example: runnemo_tgt.ksh -t $mypath -A mycomp - j $traj -v nemotam -k $nopert -f $diag

- non perturbated outputs is be saved into $mypath/$nopert directory
- final diagnostic outputs is saved into $mypath/$diag directory

## Requierements

To run the some adjoint test we need the following features:

- a direct trajectory (computed with NEMO)
- additionnal namelist to rule loop number and actual
**p**- namtst_tlm (tangent test parameter: see example in setup_tam_orca2)

- runnemo_tgt.ksh script uses two short python codes (filesmove.py and diagmerge.py) to save intermediate outputs and build-up the final outputs.

## Outputs

The outputs, located in $mypath/$diag/ directory, is called **tan_diag.global_0000**.

## Test Pass Criteria

With the above notations, we focus on Ep and Rp.

## Results

All subroutines tested both criteria except for:

**dynspg_flt**: fails on Rp criteria, root cause under investigation**traadv_cen2**: fails on the Rp criteria due to tangent approximation (not considering the volume finite part of the scheme).