I am fairly certain what people are telling you is "noise" are actually JPEG compression artifacts. Unlike RAW images, JPEG images use a form of lossy compression...that means that some degree of detail and perfection in an image is permanently lost when saving a JPEG image. From what I can tell, the photo you posted is not noisy at all, however it does display a fair amount of mild JPEG artifacting and a little bit of banding (usually called posterization) in the bottom portion.
This is a pretty normal consequence of JPEG compression with low-contrast scenes and smooth gradients. JPEG compression is excellent when there is a lot of detail, however it tends to degrade quality pretty quickly when there is a minimal amount of detail, and does quite poorly with smooth gradients. Your photo has both a low amount of detail (by design...its a photo of fog, after all), and has a fairly smooth gradient from top to bottom. You may have been able to avoid the artifacts and posterization if you had used a lower level of compression (higher JPEG "quality"), but to fully eliminate all the negatives, you would have had to have used one of the top two JPEG quality settings, which pretty much eliminates any space savings gains.
If you want to avoid the quality issues with JPEG, you should use RAW if you can. Whenever you save a JPEG, from RAW or anything else, you should try to keep the quality setting around 70-80 (or "high"), as that will usually produce some useful space saving without degrading quality too much.