Nothin but dust, baby.
Phytoplankton blooms are complex beasts. We've got a general idea of what causes them, and a general idea of what collapses them, but the intricacies are hard to capture. This is going to be a frame-challenge type answer; if you're not excited about those, you've been warned. If you're excited about the little green things, read on.
Dust storms aren't regular, annual events
The question as asked conflates two phenomena connected to phytoplankton blooms - one, that blooms appear and decay with regularity; and two, that blooms are sometimes caused by dust storms. Both of these are well documented, but they're entirely separate events.
Phytoplankton blooms have several regular, periodic cycles that they go through. These are best outlined by the recent review here, but I'll do my best to summarize the process and explain why they've got a seasonal behavior.
These little green things thrive on two things: nutrients and sunlight. In the ocean, sunlight is abundant right near the surface; but nutrients are more easily found at depth.
Side comment: one other answer noted that "nutrients tend to sink", which isn't quite true. It certainly looks that way, if you look at element profiles in the ocean, especially for common nutrients like P, N, or Fe. However, these nutrients are instead found in low concentrations at the surface because they're all tied up in cell structures rather than dissolved in the water column. When dead organisms sink in the water column, those elements are remineralized and returned to their dissolved form.
Phytoplankton deal with this discrepancy by forming something known as the Deep Chlorophyll Maximum (DCM) at a depth deep enough to have nutrients while avoiding missing out on the sunlight from above. When nutrients can be found closer to the surface, the phytoplankton also tend to get shallower.
So, seasonality. The ocean is normally thermally stratified - organized vertically by density - which makes it difficult for deep water (full of nutrients) and shallow water (full of sunlight) to mix. However, this stratification breaks down when the surface water is about the same temperature as the deeper water. This happens in the winter - check out the graphic below, from this excellent website:
Of course, during the winter there's not a whole of sunlight - so when the sun comes back in the spring, we get a phytoplankton bloom that happens over an entire hemisphere! Check out the link here for an excellent animation from the NASA Earth Observatory demonstrating this. That's the main annual cycle.
In lakes, we often get two blooms - one in spring and one in fall. The logic above still applies, but in lakes we get reverse stratification in the winter that means that mixing happens more in spring and fall than in winter. A similar graphic from Nat Geo below explains this better:
This means that lakes often have semiannual blooms, in both spring and fall.
The final kind of blooms we get in the ocean aren't regular at all. That's because these blooms aren't triggered by any kind of seasonal or annual cycle (especially if we're not including fertilizer runoff in the growing season), but are instead triggered by random events such as dust storms, volcanoes, or rivers.
Dust storms are especially powerful fertilizers because the ocean as a whole isn't limited by a single nutrient. Some areas desperately need phosphorus and have nitrogen to spare, while other areas are iron-limited but otherwise high-nutrient. The figure below summarizes this for diatoms, a major constituent of phytoplankton (from this paper):
Dust is great because it's full of all these things! Most notably, the iron and phosphate nutrient cycles have no airborne component, meaning that you can't pull those nutrients out of thin air like you can with nitrogen. Dust is generally made of rocks (or, rocks are made of dust?), so this is one way to transport a whole lot of phosphate and iron into the surface ocean. Rather than rising up from the depths, it's being dropped from above. That makes it really hard to mimic, especially on a global scale.
Dust-triggered blooms are unique events. They're not periodic, but they are special because the transport nutrients that would normally be very hard to find in the middle of the ocean. Given that periodic blooms happen to at most a hemisphere at a time (i.e. not global), and that dust-triggered blooms aren't annual, a frame-challenge answer to this question is necessary.