Press ~ to open the console, and use the command FW to force weather.  Please note that if the region you are in does not have the weather you are testing in its list, it will revert to a different weather in a couple of seconds.  That's not a bug that's just how the engine works.

fw xxx 

where xxx = the numeric ID listed below.  For example, fw c8220 will trigger a rain storm.

LIST OF VANILLA WEATHERS ALTERED / EDITED BY TRUE STORMS

SkyrimFog [WTHR:000C821E]
SkyrimOvercastRain [WTHR:000C821F]
SkyrimStormRain [WTHR:000C8220]
SkyrimStormSnow [WTHR:000C8221]
FXMagicStormRain [WTHR:000D4886]
SkyrimOvercastRainMA [WTHR:0010A231]
SkyrimFogMA [WTHR:0010A232]
SkyrimFogCO [WTHR:0010A235]
SkyrimOvercastRainRE [WTHR:0010A238]
SkyrimFogRE [WTHR:0010A239]
SkyrimStormRainFF [WTHR:0010A23C]
SkyrimOvercastRainFF [WTHR:0010A23D]
SkyrimFogFF [WTHR:0010A23E]
SkyrimStormRainTU [WTHR:0010A241]
SkyrimOvercastRainTU [WTHR:0010A242]
SkyrimOvercastRainVT [WTHR:0010A7A6]
SkyrimFogVT [WTHR:0010A7A7]
RiftenOvercastFog [WTHR:0010FE7E]

NEW WEATHERS / IDs -- FOR THE FOLLOWING WEATHERS, REPLACE 04 WITH THE POSITION IN YOUR LOAD ORDER!

TS_SkyrimStormRainRE [WTHR:04001318]
TS_SolstheimDustStorm [WTHR:04001323]
TS_SolstheimBurningAshfall [WTHR:0400132D]
TS_SolstheimAshenFog [WTHR:0400132F]
TS_SolstheimRainstorm [WTHR:04001330]