Meech sits in sixth overall after a fourth and 13th in the two races, with fellow Tauranga sailors Tom Saunders (26th) and Andrew McKenzie (28th) having disappointing starts to the regatta.
Thompson's day illustrated what could happen at Gamagori, when he followed up a second placing in the first race with a 42nd in the other race to sit 18th overall.
Meech said the wind shifted quickly and was quite up and down.
"At times you were sitting in your boat praying for some breeze. Luckily for me, I had two OK races so I'm happy with how I sailed today. We actually did three races today but the first was abandoned after we got most of the way around the track. It just glassed off and there was no wind so, unfortunately, they had to call that one," he said.
"Everyone thought that was it for the day but, as we started to tow in, the wind filled in and so we ended up going back out on to the course and got two reasonable races. The forecast is looking like more of the same for the rest of the week so it looks like we are going to be rained on every day. But it could be worse."
Isaac McHardie and William McKenzie dealt with the conditions the best in the 49er fleet and won the only race of the day, finishing seven seconds ahead of world championship silver medallists James Peters and Fynn Sterritt of Great Britain.
Logan Dunning Beck and Oscar Gunn (sixth) and Josh Porebski and Tauranga's Trent Rippey (10th) are handily placed in the 20-boat fleet.
The men's 470 combination of Paul Snow-Hansen and Dan Willcox started well with a second but followed it up with a 22nd to sit 11th overall and Susannah Pyatt was 14th in the only race possible in the women's Laser Radial.