  Electrostatic (E) fields associated with the interaction of a
well-controlled, high-power, nanosecond laser pulse with an underdense plasma
are diagnosed by proton radiography. Using a current 3D wave propagation code
equipped with nonlinear and nonlocal hydrodynamics, we can model the measured
E-fields that are driven by the laser ponderomotive force in the region where
the laser undergoes filamentation. However, strong fields of up to 110 MV/m
measured in the first millimeter of propagation cannot be reproduced in the
simulations. This could point to the presence of unexpected strong thermal
electron pressure gradients possibly linked to ion acoustic turbulence, thus
emphasizing the need for the development of full kinetic collisional
simulations in order to properly model laser-plasma interaction in these
strongly nonlinear conditions.
