Sorry for the delay - my internet connection has been down this evening.
As Koz said, Audacity processes internally at 32 bit. When you Export the audio as a 16 bit file, dither is added. The dither noise is introduced when down-sampling from 32 bit to 16 bit. Importing a 16 bit file or up-sampling do not introduce any dither noise.
You will notice that the noise is above 15kHz - that is because the type of dither you are using is “shaped” to push the noise where it will be least noticeable while still performing its necessary function.
Other types of dither produce different responses, for example “triangular” produces a lower level of noise, but across the full audio spectrum.
Setting the dither to “rectangle” will introduce virtually no noise when down-sampling a silent track, however the dither is likely to be less effective and more noisy on a non-silent track.
“Triangle” or “shaped” dither are usually considered to be best for music. My preference is for shaped.
To avoid the dither noise you can switch dither off in Preferences (not recommended), or Export as 32 bit audio. To Export 32 bit audio you need to select “other uncompressed files” as the Export file type, then in “Options” you can select a 32 bit format (such as 32 bit signed Microsoft WAV).
It’s curious to think that virtually every CD of the last 20 years has dither noise present (though usually not in the silence between tracks).