Context : Red giant stars are important tracers of stellar populations in the Galaxy and beyond , thus accurate modeling of their structure and related observable properties is of great importance .
Three-dimensional ( 3D ) hydrodynamical stellar atmosphere models offer a new level of realism in the modeling of red giant atmospheres but still need to be established as standard tools .

Aims : We investigate the character and role of convection in the atmosphere of a prototypical red giant located close to the red giant branch ( RGB ) tip with atmospheric parameters , T _ { eff } = 3660 K , \log g = 1.0 , [ \mathrm { M / H } ] = 0.0 .

Methods : Differential analysis of the atmospheric structures is performed using the 3D hydrodynamical and 1D classical atmosphere models calculated with the CO ^ { 5 } BOLD and LHD codes , respectively .
All models share identical atmospheric parameters , elemental composition , opacities and equation-of-state .

Results : We find that the atmosphere of this particular red giant consists of two rather distinct regions : the lower atmosphere dominated by convective motions and the upper atmosphere dominated by wave activity .
Convective motions form a prominent granulation pattern with an intensity contrast ( \sim 18 \% ) which is larger than in the solar models ( \sim 15 \% ) .
The upper atmosphere is frequently traversed by fast shock waves , with vertical and horizontal velocities of up to Mach \sim 2.5 and \sim 6.0 , respectively .
The typical diameter of the granules amounts to \sim 5 Gm which translates into \sim 400 granules covering the whole stellar surface .
The turbulent pressure in the giant model contributes up to \sim 35 \% to the total ( i.e. , gas plus turbulent ) pressure which shows that it can not be neglected in stellar atmosphere and evolutionary modeling .
However , there exists no combination of the mixing-length parameter , \alpha _ { \mathrm { MLT } } , and turbulent pressure , P _ { turb } , that would allow to satisfactorily reproduce the 3D temperature-pressure profile with 1D atmosphere models based on a standard formulation of mixing-length theory .

Conclusions :