Hi, and welcome to composting. Although we do 90% of our composting in piles on the ground, we also have a compostumbler that a neighbor gave us.
I don't think the 2-to-1 ratio is correct. If you compost on the ground it's about 4-to-1, brown to green (or nitrogen).
Your location does present a potential problem in terms of getting brown stuff. Do you have access to anyone who does woodworking so you could get the sawdust from them? Or can you get tree trimmings from a service that cleans up when branches fall in the cold weather? If getting dried leaves aren't an option, you're going to have to think in terms of getting some of the other "browns".
Dirt Doctor lists these things as options: leaves, hay, grass clippings, tree trimmings, bark, and sawdust. I'm guessing you could at least get bark, tree trimmings and/or sawdust.
The other thing to do -- if you have the space -- is to collect leaves when they do
fall and get yourself a BIG pile or bin of them so you can use them in your tumbler when dried leaves are scarce.
We have two very large enclosures we use to collect leaves. We pick up dozens upon dozens of bags in the fall and early winter, as well as cleaning up the yards of neighbors on our street to get their leaves and pine straw, and sweeping up the street itself to get the leaves and pine straw that collects all along the curbs. (The neighbors LOVE us!
Also, you should turn your tumbler -- one turn -- every day. When we started doing that, it really helped. We got finished compost in a little over a month.
Yes, you can add kitchen scraps, but think of them in terms of green and brown and don't put too much in, or add anymore after the first week or so. If you do, your "pile" inside your tumbler will never finish.
Oh, and make sure the thing is jam-packed FULL when you start. To get a pile (on the ground) cooking it has to be 3' x 3' x 3'. Your bin -- if it's like ours -- is constructed so that you get 3' x 3' x 3' inside the cylinder. But it has to be FULL to cook.
Make sure your pile (tumbler contents) stay moist. Look inside a couple of times a week and see if it's moist. If not, add a little water. It should never smell bad.
One final note, lots of people add meat and/or dairy to their piles. We do not. And I don't put it in the tumbler either.
Once we know how to get a pile hot and KEEP it hot for a couple of days, then I will start putting in dead animal carcasses, meat, or dairy. Until then, I don't want any smell, or any vermin near our compost.
Hope that helps!
**Take time to stop and smell the flowers
(or... as my ladybug refrigerator magnet says
"take time to stop and eat