In my experience 3/8" is too thick. You will just never be happy with the results. The parameters will have to be so tight as far as electricals and travel speed that it is almost impossible keep the bead from sagging. And if the sulfur (or any element that changes wetting and/or penetration patterns) content changes all bets are off. The parameters you set will have to be adjusted
And this is with rolloouts. Forget all-position.
There may be someone out there doing it better but I couldn't do it. 1/4" is about the max.
If you are doing roll outs, and you have some money to spend I would recommend GTAW with AVC bevel slightly at the top and run two passes.

Hi Petty 4345!

For the most part, you're just about at the limit @ 3/8" thick... You may just want to check out these articles below because the limit for 304 SS is around 8 to 10mm... 3/8" = 9.525mm which is right at the limit and as Jeff articulated so eloquently from his own experience, the plasma column is a bit erratic at the upper limit of the thickness range of 10mm. ;) I would look into the Hybrid welding process of combining GMAW & PAW for your application, or even the Double Sided Arc Welding process (DSAW) and determine if they are practical towards your specific application. I almost forgot!!! "WELDCOME TO THE WORLD'S GREATEST WELDING FORUM!!! :) :) :)"

An interesting article below as well as in the following one also, although the materials welded were much thinner than what you have in mind:

http://files.aws.org/wj/supplement/Pierce/ARTICLE3.pdf

Some good information in this article:

http://files.aws.org/wj/supplement/Zhang/ARTICLE4.pdf

Here's a paper to consider ordering:

http://www.dtic.mil/srch/doc?collection=t3&id=ADA074340

If you could translate this from Japanese to English, I'm sure one could find some valuable information regarding PAW of thick plate over 10mm:

http://nels.nii.ac.jp/els/110003392285.pdf?id=ART0003943286&type=pdf&lang=en&host=cinii&order_no=&ppv_type=0&lang_sw=&no=1266094783&cp=

Here's an interesting article from the Welding Research Institute of Osaka University from  back in 1984:

http://ir.library.osaka-u.ac.jp/metadb/up/LIBJWRIK01/jwri13_01_007.pdf

This article covers a form of Hybrid welding (Combination of GMAW & PAW):

http://www.weldingsolutionsinc.com/files/WELDING2%5B1%5D.pdf

These folks do some interesting stuff:

http://www.plasma-laser.com/new/index.asp

Here's an article in their web site from the original article from the Welding journal of October 2007:

http://www.plasma-laser.com/NEW/technologies.asp?catID=2&subCatID=39

Here's the Welding Solutions Incorporated Homepage:

http://www.weldingsolutionsinc.com/home2.html

Whoever wants to develop a newer type of PAW torch which handles a Ternary gas plasma configuration for commercial applications, then you can contact this person @ NASA:

http://techtran.msfc.nasa.gov/tech_ops/plasma.pdf

This is an article from the Welding Journal from I believe November of 2002 describing "Double sided Arc Welding" or, DSAW which is another type of "hybrid" arc welding process whereby two plates are welded together from both sides simultaneously with the use of PAW on one side and GTAW on the opposite side to produce full penetration welds in deep narrow groove butt joints of thickness of up to 1/2 inch and possibly thicker with less heat input than conventional Keyhole PAW... This may be just the solution you're looking for, and I'm sure that further studies have been made regarding some of the concerns listed in the conclusions of this report from what was then an experimental welding process:

http://files.aws.org/wj/supplement/11-2002-ZHANG-s.pdf

Here's an interesting brochure from Air Liquide:

http://www.docstoc.com/docs/20504443/File-TIG-welding-and-plasma-welding

These folks claim they can weld up to 10mm thick stainless with their PAW Keyhole power source/torch system, and they're from both the UK and Denmark:

http://www.migatronic-automation.com/media/pi_plasma_uk.pdf

I personally would rather trust the word of Fronius instead:

http://www.fronius.com/cps/rde/xchg/SID-90492575-52F3920C/fronius_international/hs.xsl/79_13504_ENG_HTML.htm

Here is an interesting article for Jeff to take a look at:

http://www.modlab.lv/publications/mmp2006/pdfs/143-148.pdf

Here's another one for you Jeff, and it may be applicable for the OP's application also:

http://www.freepatentsonline.com/5225654.pdf

Respectfully,
Henry