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Phytoplankton are not to be taken for granted. Not only do they form the core of marine food webs around the world, they also release half of the world's oxygen. But phytoplankton, being plant-like organisms, need nutrients for their blooms to survive and thrive into populations large enough to be visible from space. For many, the origins of those nutrients come from one of the least likely sources: desert dust storms swept up from far away by deserts. As the dust settles down to the oceans, they drop down enough nutrients to create these vast blooms on an annual basis.

But what else could feed a plankton bloom on a global scale and an annual basis?

You could say that volcanic ash could be the answer, but there's a problem--unlike dust storms, volcanoes don't erupt at once or regularly. Two eruptions from one same volcano could be years or even decades apart, and that sort of duration gap won't do for plankton blooms. So what else could feed a plankton bloom on a global scale and an annual basis?

EDIT--NO manmade processes! Everyting MUST be natural!

Rivers.

https://earthobservatory.nasa.gov/images/1257/mississippi-river-sediment-plume

Depicted: the Mississippi dumping its load of sediment into the Gulf of Mexico. River flow is cyclical in most places, with high flow during rainy season or spring melt and low flow during dry season / winter or summer. During high flow, nutrients move from the land to the river and on to the sea. With the advent of synthetic fertilizer this can be too much of a good thing - so much nitrogen and phosphorus that they produce massive blooms, that then die.

Icebergs.

Icebergs generation is periodic, both intrayear and over longer periods.

https://www.nationalgeographic.org/media/iceberg-frequency/

Icebergs that have scraped along the land can ferry nutrients out to sea, releasing them slowly as the ice melts.

https://phys.org/news/2019-03-mystery-green-icebergs.html

The green icebergs have been a curiosity to scientists for decades,
but now glaciologists report in a new study that they suspect iron
oxides in rock dust from Antarctica's mainland are turning some
icebergs green... Iron is a key nutrient for phytoplankton,
microscopic plants that form the base of the marine food web. But iron
is scarce in many areas of the ocean.

If experiments prove the new theory right, it would mean green
icebergs are ferrying precious iron from Antarctica's mainland to the
open sea when they break off, providing this key nutrient to the
organisms that support nearly all marine life.

Eventually, phytoplankton don't really feed on dust, but on nitrogen (N), phosphorus (P), iron(Fe), and the various other nutrients plants need, that may compose it. But if a desert hold such nutrients, it will not stay a desert for long.

An example of a yearly massive bloom could be a mass migration of ground animal, on shore, for reproductive purpose (a bit like toad, that need water even if they are most the time ground animal). They will stay on the shore for some weeks, defecating and urinating, and releasing a massive dose of nitrates, phosphate and so on...

Not accepted as an answer since last edit :
Second example, industrial activity, mainly agriculture, may lead to algae bloom (like in Brittany, France, with the famous green and smelly algae). And since plants grow on yearly cycle, fertilizer are used on a yearly basis

Up-welling. Nutrients tend to sink to the bottom, or to deep water where not enough light reaches to keep photosynthetic life forms thriving. If there is some mechanism to vigorously return deep water to the surface then it can bring the nutrients with it.

https://oceanservice.noaa.gov/facts/upwelling.html

Up-welling might well be a seasonal thing. For example, currents could flow in one direction half the year when the snow melts in this hemisphere and builds up in the other. Then in the other direction for the other half of the year. This could produce a seasonal stirring of the deeper ocean layers.

Up-welling could be driven by temperature differences produced by geological heating that does not rise to the level of volcanoes. Water is at its highest density at close to 1.5°C. So if you have something that warms the depths it will bring the deep water back to the surface. This probably isn't seasonal.

In exotic places with exotic tides, that might do it. If a large moon had an exceedingly eccentric orbit, you could have extreme tides for the portion of the moon's orbit when it was closest, then far weaker tides the rest of the time.

Seasonal wind patterns might also do it. Wind tends to push the surface water along more than the deep water, so it tends to arrive at the surface near the down-wind land. Then it tends to sink and return to the depths. In reverse at the up-wind land, the water tends to get pushed away from the land on the surface, pulling the deep water up.

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.

Periodic blooms

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.

Irregular blooms

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.

TL;DR:

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.