Atmospheric rivers are corridors of strong atmospheric water vapor transport. Such transport is dependent on both the water vapor content of the atmosphere and the strength of the flow. Values are typically highest when the water vapor content of the atmosphere is high and the flow is strong.
Atmospheric rivers can be associated with heavy precipitation, but strong atmospheric water vapor transport, by itself, doesn't generate precipitation. Some forcing is needed to lift the airmass, form clouds, and generate precipitation.
In the case of an atmospheric river, one possibility is to have it cross a mountain barrier, which yields strong upslope flow and precipitation enhancement. This happens commonly in the mountains of California during atmospheric river landfall.
Another option is to have large-scale forcing, such as the ascent typically found near cold fronts or along warm fronts.
With this in mind, the plot below shows a time-height section (time in this case increasing to the right) at a location near Salt Lake Cit airport from the 0600 UTC initialize GFS. The color fill is the water vapor transport, and you can see how it maximizes at around 0000 UTC 23 March (6 PM MDT Thursday). Note, however, that at this location, although there is a peak in precipitation at that time, a greater peak is found later, when the vapor fluxes are lower, but when the surface-based cold front is moving through.
That forecast above, however, is from a model (GFS) with relatively flat terrain. If one were to go to a places like Snowbasin or Sundance, the story could be different. At these locations, southwesterly flow is oriented strongly across the local topography and and significant generation of precipitation can occur.
Although I hesitate to use the actual totals produced by the 3-km NAM, the forecast below valid 0300 UTC 23 March (9 PM MDT Thursday) illustrates this well. Note the heavier precipitation in the areas around Mt. Timpanogos (Sundance) and the northern Wasatch (Snowbasin), as well as over the Uintas. These are areas where the mountains are oriented across the crest-level (10,000 ft) flow. You can also see the heavier precipitation over northwest Utah where there is forcing along the cold front.
Sadly, ahead of the cold front, this is a very warm storm. Our NAM derived forecast for the upper Cottonwoods shows wet bulb zero levels (typically the snow level is about 1000 feet below this) reaching as high as 10000 feet Thursday before lowering late Thursday night/early Friday morning with the frontal passage. Much of the precipitation ahead of the front will fall in the form of rain at elevations below 8000-9000 feet.
Keep an eye on official forecasts the next couple of days. Ultimately, the timing, intensity, and amount of precipitation will depend on both the characteristics of the atmospheric river, the mountain effects, and the cold front. I will also add that there is the potential for strong pre-frontal souhterly winds and thunderstorms late Thursday and Thursday night. Too much to cover in a blog post from Seattle.