Context : The treatment of mixing processes is still one of the major uncertainties in 1D stellar evolution models .
This is mostly due to the need to parametrize and approximate aspects of hydrodynamics in hydrostatic codes .
In particular , the effect of hydrodynamic instabilities in rotating stars , for example , dynamical shear instability , evades consistent description .

Aims : We intend to study the accuracy of the diffusion approximation to dynamical shear in hydrostatic stellar evolution models by comparing 1D models to a first-principle hydrodynamics simulation starting from the same initial conditions .

Methods : We chose an initial model calculated with the stellar evolution code GENEC that is just at the onset of a dynamical shear instability but does not show any other instabilities ( e.g. , convection ) .
This was mapped to the hydrodynamics code SLH to perform a 2D simulation in the equatorial plane .
We compare the resulting profiles in the two codes and compute an effective diffusion coefficient for the hydro simulation .

Results : Shear instabilities develop in the 2D simulation in the regions predicted by linear theory to become unstable in the 1D stellar evolution model .
Angular velocity and chemical composition is redistributed in the unstable region , thereby creating new unstable regions .
After a period of time , the system settles in a symmetric , steady state , which is Richardson stable everywhere in the 2D simulation , whereas the instability remains for longer in the 1D model due to the limitations of the current implementation in the 1D code .
A spatially resolved diffusion coefficient is extracted by comparing the initial and final profiles of mean atomic mass .

Conclusions : The presented simulation gives a first insight on hydrodynamics of shear instabilities in a real stellar environment and even allows us to directly extract an effective diffusion coefficient .
We see evidence for a critical Richardson number of 0.25 as regions above this threshold remain stable for the course of the simulation .