Francis Alÿs is running across a field, chasing a tornado that whirls and whips and plumes into the clear Mexican sky. It is a small tornado, it must be said, but big enough to engulf trees, and any passing Belgian that gets in its path: a storm of dust, stones, clods of earth, straw and animal droppings. The artist pants across the dry earth with his handheld camera and steps right into the the thunderous sizzle and roar, the no-visibility brown-out. Inside the vortex there is a sudden momentary stillness, as he stands in a column of dead air, before being pelted once again by the infernal dust. Time and again he races towards these high-altitude tornadoes before disappearing.  Tornado is an almost hour-long video, as gripping as it is terrifying. Here comes another one; there he goes again. Eventually, we glimpse the artist, bruised and gulping for air, prone in the dust. Filmed over 10 years, it is the latest and most surprising work in a new show that focuses on the Antwerp-born artist's actions and performances.  Alÿs never struck me as a brave guy. He is probably best known in the UK for the exhibition he mounted at the National Portrait Gallery last year, of over 200 amateur portraits of Saint Fabiola, collected from auctions and thrift stores, all based on a lost 19th-century original by Jean-Jacques Henner. The collection is still touring the world. On another occasion, Alÿs released a wild fox into the same gallery, to lope and wander through the rooms at night, followed by CCTV security cameras.  Like a fox, Alÿs does a lot of walking and standing about and watching, making understated films and videos, as well as genial little paintings and jerky, hand-drawn animations. Tate Modern's exhibition is a delight: thought-provoking, funny and full of pathos, as well as danger. I have seen other survey shows of his work, but none as good as this. It begins with a film of a mirage quivering on a Patagonian road and ends on a similarly quiet note, with a short animation of a woman pouring liquid from one glass to another, and back again, to a quietly lilting song. In between, there are marching bands, the wail of police sirens, shouts and silences, like that stillness at the centre of the tornado. One room is entirely dedicated to silence, with rubber tiles deadening the sound of one's footfalls, each tile decorated with a picture of a finger raised to the lips, like a sign in some old-fashioned library.  Alÿs has lived in Mexico City since the mid 1990s, and many of his works take place in the city's teeming streets and around the Plaza de la Constitución, a huge paved piazza with a very tall flagpole at its centre. He has shepherded sheep around this flagpole, to the sound of church bells; he has filmed the city's itinerant workers queuing in its shadow, as it sweeps like a sundial through the long day. He has even queued with the artisan plumbers, electricians and housepainters who stand there with their painted signs, offering their services for hire. Alÿs had his own sign: Tourist, it read.  What can a tourist offer apart from an outsider's bewilderment, a guidebook's secondhand knowledge, ignorance and cultural misunderstanding? A tourist can always go home, but Alÿs decided to stay in Mexico City for decades. Once, he pushed a large block of ice through the streets, skidding and sliding till all that was left was a dirty pebble melting on the greasy tarmac. Sweating through the streets, you think he might evaporate, too, and that what we are watching is a man performing his own insignificance and futility. But I also imagine his persistence as a walking anecdote: "Did you see that stupid gringo, pushing that block of ice?" Alÿs is a perpetual gringo, whether walking through Mexico, running a stick across the railings in front of posh London houses, or trailing a line of green paint from a leaking can through divided Jerusalem, following the invisible Green Line.  Another time, Alÿs walked into a Mexican gun shop and bought himself a pistol, cocked it and carried it in full view through the streets, walking fast, the gun aimed at the sidewalk. He looks like a man intent on some murderous purpose. In the end, the cops roar up and he is bundled into the patrol car. We watch this performance – if performance it is – twice, on two adjacent screens, the action on the righthand screen emblazoned with the word re-enactment. Alÿs, one decides, is foolhardy rather than brave. He walks a fine line between public nuisance and menace, holy fool and extremely annoying person. There is something about his height (he is well over 2m tall), his gangling lope and faux-naive innocence that not only allows him to get away with such acts, but also to persuade others to go along with his schemes. In 2002, he persuaded several hundred students and locals to arm themselves with shovels and attempt to move a huge sand dune on the outskirts of Lima, Peru; they shuffled in line over the dune, shovelling the sand in front of them as they went. Of course, it didn't work.  There's more to When Faith Moves Mountains than mere absurdity. All that effort was an attempt at collectivity, and meant to highlight the possibility of co-ordinating some kind of urbanism among the dunes and the uncontrolled, ramshackle, ad-hoc housing developments around the Peruvian capital. People do daft things all the time, but it doesn't often make for meaningful art. In Alÿs's case, it does. He once crossed the border between Mexico and the US by going the long way round – an entire Phileas Fogg-style circumnavigation, in order to bypass the border crossing of a few heavily policed yards. He recently returned from Kabul, where he took part in a UN-sponsored NGO mission to decide what to do about the Bamiyan Buddhas, the pre-Islamic Buddhist monuments blown up by the Taliban in 2001. The world is full of painful absurdities, and in his art Alÿs celebrates hope and the power of the imagination, in the face of misery and iniquity.  Here comes the mariachi band, with farting trombones, wailing trumpets and clashing cymbals, sweatily orchestrating Alÿs's repeated attempts to drive a VW beetle up a steep unmade road to the brow of a hill, somewhere in Tijuana. The band strikes up, gathers itself, sets off full of hope and bravura, and as the car runs out of steam so does the band. The music wilts, dies and then the car sets off again, full of purpose, going nowhere. Weirdly, hilariously, it is an inspiration.
Artist Francis Alÿs has poured paint through Israeli border controls, leapt into tornadoes and attempted to move mountains. Is he brave or stupid? Adrian Searle finds out in an inspiring new show of his work