Are you sure it is Caulerpa? It looks like Bryopsis. Have you seen the stolons? In any case, I think it isn't Caulerpa longifolia. You can also check for Caulerpa sertularioides...
I have collected this specimen along the coast of karaikal and the image which i provided is a microscopic image at a resolution of 5X. The ramuli was not starting from the base, so it wont be Caulerpa sp.. The ramuli size and structure characteristics was matching with the Bryopsis sp. In India Bryopsis plumosa and B.pennata was reported species, therefore the collected specimen may be B.plumosa or B.pennata. Help me to confirm it.
Based onthe assimilatory frond we may not come to conclusion, if the TS of the stolon showing the trabaeculae or skeletal strands then it may be Caulerpa species.
In our coast we are having Bryopsis pennata and Pseduo byropsis mucronata, but above figs are not tally with them
Please upload the entire thallus photographs with habitat.
Thank you Dr Rajan for the pictures, It is look like Bryopsis based on the habit but I am not sure. If the TS of Rhizome/stolon shows the trabaeculae then it may be Caulerpa. You please do this work for proper identification.