First of all lets see the shape of the dome:
That's 40km in radius ( (2+2)*10 = 40km ) and 3km in height ( 0.3*10 = 3km )
many thanks to Liath for revealing this online gem
That's pretty flat. I used Kelly Thomas's approach and compared it to a geological feature, but a hill instead of a mountain since this is a lot less steep. However, it turns out the tallest hills are about 1km tall and still cause rain shadows despite the flatness. Here's some examples:
The Judaean Mountains which are 1km tall and relatively flat compared to what we think of as mountains, but cause enough of a difference to shadow the Judaean Desert and the Dead Sea. Now that's an effect.
The Tibetan Plateau, which again, while quite tall, is a relatively flat area, not very jagged and steep. At an average elevation of 4.5km, it creates a steppe region and funnels water down into lakes. This gives us a sense of scale - your dome is a huge mountain.
I apparently followed Kelly's footsteps again because I also found the Alaskan woody tundra mentioned in the answer and it seems to indeed be caused by rain shadowing. The question is, if it is already a tundra and then you add a rain shadow, what happens?
Looking around for a mountainless tundra, I found nothing - in fact searching for it just gave me results for mountains related to tundras. However, the arctic tundra variant that applies to the poles might be closer. The problem is, the climate there is so cold, it's like what do you expect? They also seem to have an altitude to them (maybe your area is a plateau/plains at high altitude). This page however implies that the antarctic tundra isn't really a tundra but more like a desert that is too damn cold to not be a tundra-like biome.
The problem remains - unless your dome caused the tundra in the first place - it would be cold plains or a regular desert (just not sandy). But tundra is believable enough I guess.
As Vincent mentions, you need to recycle water and air, but considering how large the dome is and how mountains in tundras tend to be the sources of rivers and lakes, why not just collect the water off the dome and create an artificial river or lake inside the dome? Best park evar. With enough rainfall, you won't need to dig for water, it'll just fall out of the sky.
Here's the extra issue with the conditions within the dome. The outside will probably not be too much affected by your dome, considering it's already practically a desert. However, your dome is large enough to have its own weather :D
Take a look at this:
There's more of these I assume, but we have important information from this: the enclosed spaces are nowhere near as large as the space your dome encloses. Since its maximum depth is about 450m, we can assume the areas inside are going to be less tall and your dome is 6 times taller (or 4 if it's 2km). Since you're in a tundra it's cold enough to have vapors condense. Also, it's as long as your city but still a cave - so your city will have more enclosed space that that entire beast of a cave in total - I'd assume about 10-20 times as much, all in one area.
The question is, does sunlight change this? I'd assume not since clouds generally don't seem to care about sunlight, nor does the cave, but I'm no meteorologist. It may cause a warmer climate inside though. You could have a greenhouse effect on your hands, but the temperature (I assume) shouldn't be too high. A lot of humidity and heat will be trapped due to the people (assuming they're warmblooded) and water, but you need some filtration and conditioning for the air anyway, just in case, so that might be able to regulate things. Considering you'll have weather though, it might (again, I assume) not be as critical since there's plenty of space for circulation. Also, you might want to expect birds :P
At 10x40km (400 km^2) and considering this and this we can see that the New York metropolitan area covers about 34km^2 and your city is over 10 times as big. That's pretty big and I'd assume that its microclimate will be closer to an actual climate, who's nature will depend heavily on what activities take place within.
An interesting aside - the buckyball guy had proposed a dome for New York back in the 60s.