On Tuesday, September 8, 2015 the archdiocese of Los Angeles (largest in the USA with some 5 million Catholics in 280 parishes) received three new auxiliary bishops. The Most Rev. Joseph Brennan, The Most Rev. David O’Connell both priests of the archdiocese and The Most Rev. Robert Barron, a priest of the archdiocese of Chicago were ordained by The Most Rev. José Gomez, archbishop of Los Angeles. The coats of arms of Bishops Brennan, O’Connell and Barron are below. All three were designed by James Noonan and depicted by Linda Nicholson.
EXARANDORUM 
I quite like the use of the chief of the diocese, but wonder if it would merely be dropped if they were to be “upgraded” to being the Ordinary in a diocese. In the case of bishop Barron, with the chief per wavy, it’s not completely uncomplicated.
It is not a chief “of the archdiocese”. Rather, the chief on all three contains charges which allude to the archdiocese of Los Angeles. This is perfectly acceptable. Even if any of them is promoted (the correct word) to be a diocesan and translated to another See this would remain as a reminder they once served in LA. In addition, it is always best to remember that when assuming arms a bishop is not supposed to think about any heraldic considerations”if” they are later promoted. While it certainly happens, bishops are to proceed with the understanding that unless required they will not be moved.
Ah! I see! I made a hasty assumption, which apparently was incorrect. I thought it was a rather clever way of indicating which diocese they serve, but did not see it as a permanent fixture of the arms.
By the way: Is it correct for bishop Barron to display his membership in the Order of the Holy Sepulchre in that way? Shouldn’t the arms be placed upon the cross instead?
It is correct because it corresponds to the rank in the EOHS he currently holds. If he is promoted in that order (very likely now that he is a bishop) he may place his arms on the cross of the Order.
Thank you! I was unaware that the EOHS marked specific ranks in that way.
How does one blazon the “white bit” in the second arms, please?
You don’t. The entire field below the chief is blazoned and then the chief is blazoned. So, in blazoning the whole field one would say, “Tierced in pairle Argent, Azure and Vert…” That is to say you describe how the shield is divided into three sections and name the tinctures of the field of each section starting with the top center, then the dexter base and finally the sinister base. So, the “Argent” mentioned in blazoning the tierce is the “white bit”.