A sea mist hung over the mountains of Svalbard. Under the bows of our Zodiac inflatable the water was black as ink as we nudged our way through the pack ice, and it was here, at a place called Isbukta – Ice Bay – on a day when the whole world seemed locked in a single frozen silence, that I found my first polar bear.

“Look for something yellow,” Martin Gray had said when I asked him about bear-spotting. “Then see if it moves.” Gray is a guide on the Russian research vessel that was to be my home for the next 10 days, and 18 seasons in the Arctic had given him a profound knowledge of these charismatic carnivores.

What I had seen through my binoculars was yellow, all right. It was the pale yellow of a labrador and, although it was perhaps a mile off, it stood out clearly against the snow.

And now it was walking. “Bear,” I yelled, overcome by a rush of pure adrenaline. It was an extraordinary encounter, made all the more vivid because, apart from ourselves, the bear was the only moving thing, as if all life in that vast and empty landscape was concentrated in this solitary prowling figure.

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BACKGROUND

• Afloat on an Arctic cruise adventure

• The growth of Arctic cruising

• The melting Arctic

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As we drew nearer, I could see the bloody carcass of the seal on which it had been feeding. By now it had reached the sea, where it paused for a moment to fix us with a baleful stare. Then, oblivious to the bitter cold, it plunged in and paddled away through the bobbing floes. Who could forget their first sight of the ice bear?

Armed with claws five inches long, it is the scariest predator on earth; the floe-stepper, the ghost of the tundra, sudden death in a pale fur coat; and for travellers it is one of nature’s must-see spectaculars, right up alongside the Bengal tiger and the Serengeti wildebeest migration.

Today, as climate-change tourism takes off, the Arctic is the latest wilderness to become an adventure playground, and Svalbard – a barren archipelago twice the size of Belgium – is booming. Making the most of round-the-clock daylight in the brief Arctic summer, more than 40,000 visitors flock to its shores each year, all hoping to see polar bears venturing onto the sea ice to hunt seals.

If you want to see these lords of the Arctic, my advice is not to leave it too long, because the sea ice is breaking up earlier every year and this is bad news for bears. Already they are producing fewer cubs, becoming thinner and are increasingly driven to cannibalism. “The polar bear may well become the iconic symbol of global warming,” says the Royal Geographical Society.

SVALBARD lies deep inside the Arctic Circle, only 620 miles from the North Pole, and was discovered in 1596 by William Barents, the Dutch navigator. He called it Spitsbergen, and only when the Norwegians were granted full sovereignty in 1925 did it become known as Svalbard – the Cold Coast – in recognition of the Vikings, who may have reached it in the 12th century.

Today, more than half of Svalbard is protected, a sanctuary for walruses, vast sea-bird colonies – and 3,000 bears. Despite its ethereal beauty it is also a hostile and unforgiving place. Last August, 17 British tourists were injured when a calving glacier created a miniature tsunami that almost capsized their ship, and polar bears have killed half a dozen people since 1974.

Maybe that is why Gray’s briefing on day one is the most chilling preamble to any big-game experience I have come across. “As far as polar bears are concerned,” he tells us, “you are all just vertical Gore-Texed seals.”

Gray, an Orkneyman and one-time crofter, is a true man of the north, an unreconstructed 21st-century Viking with hair like hoarfrost and ice-blue eyes that never quite conceal a laconic sense of humour. Once, unwillingly, he had to fly down to London and didn’t think much of it.

“Not my kind of tundra,” he complained. You can laugh at his fisherman’s superstitions. He will never drive his Zodiac in an anticlockwise direction. “It’s disrespectful to the sun,” he says. When it comes to polar bears, however, you know you are listening to one of the best guides in the business. “Make no mistake,” he continues, “bears don’t bluff. They don’t mock-charge. They just come and kill you. That is why I don’t see a bear 30 metres away. What I see is 5½ seconds. That’s all it takes before it’s right there in your face. And, even if you spot a bear a mile off, that means it can reach you in six minutes.”

Fortunately for us, the Akademik Sergey Vavilov is as secure as the Kremlin and we can sleep safely at night even in the midst of the sea ice. She’s a splendid ship, built in Finland for the Russians and reborn for a life of polar cruising. Not that Paul Goldstein, our expedition leader, would thank me for calling it a cruise ship.

“We don’t use the c-word here,” he says. “Rigid day-to-day itineraries? Forget it. We’re on a freewheeling mission to show you the best of Svalbard: icebergs, sea birds and, above all, bears.” Despite Goldstein’s protestations, life on board the Sergey V is as decadently western as you could wish. The decor may be clinically functional, but the cabins are snug; every lunch and dinner kicks off with a delicious tureen of heart-warming soup; and the Russian crew are unfailingly polite.

Furthermore, not only is she the quietest ship in the Arctic, she is also blessed in having Captain Valeriy Beluga in command. From his high-tech eyrie on the bridge, I watch as he guides her through the ice, as soundlessly as a seal, to within metres of a polar bear family resting on a floe. “I’ve sailed with a lot of Russian skippers in my time,” whispers Gray, “but this guy is the best by a mile”.

The bears, a mother with twin cubs, seem totally unconcerned at our close presence. The cubs are young and full of play. One rolls on its back, placing all four paws together, while the other sits upright like a child’s teddy. “All it needs is Goldilocks,” says someone as a barrage of cameras captures the moment.

Most days there are expeditions, where, trussed up like Michelin men in waterproofs, life jackets and wellington boots, we are ferried ashore by Zodiac. When we hit the beach it’s always a wet landing – hence the wellies – and the first off the boat is a guide with a gun in case there are bears about.

In Svalbard, the ice age has never gone away. Wherever you look the earth is still under construction, a desolation of frost-shattered peaks, like the shearing molars of a polar bear, and giant glaciers crawling through valleys of shale and ironstone. At first, what you see is a monochrome world of snow and rock and gunmetal seas. But in a little while you become aware of other, more subtle colours: the sepia tones of reindeer moss, the turquoise gleam of sunlit ice floes. And yet, in the midst of it all, there is life.

Shellbursts of guillemots pour overhead, riding the wild wind on the way from their fishing grounds. Out in the tideways, humpback whales surface only a Zodiac’s length away as they pursue the migrating shoals of capelin; and on the tundra, where we are dive-bombed by nesting skuas, pincushion clumps of Arctic saxifrage demonstrate the inexhaustible tenacity of nature.

One morning, we trudge inland to see the huge breeding colonies of little auks that transform the dizzy slopes of the Hornsundfjord into shrieking high-rise tenements. Beneath the cliffs, reindeer with antlers gift-wrapped in velvet are browsing in the company of barnacle geese; but it’s the auks that catch the eye.

“In Svalbard we count them in telephone numbers,” says Gray, and whenever a predatory gull appears, they erupt in panic-stricken clouds that darken the sky.

At Bourbonhamna we pick our way along a tide-line littered with Siberian driftwood and find a dead sperm whale washed up on the beach. It has been there a year, and is now little more than a mound of blubber on which ivory gulls are feeding.

Elsewhere, we creep up on a 50-strong herd of walruses, close enough to catch a whiff of their fish-factory smell as they lie packed together like rush-hour passengers on a Tube train. Apparently, walruses are thigmotactic – touchy-feely to you and me – but their eating habits are less endearing.

They feed on bivalves, but don’t bother to open them. Instead, they simply extract the flesh with their Dyno-Rod lips and spit out the shells. Sometimes, instead of landing, we just go for a Zodiac cruise around the fjords, with the pack ice crackling all around us like Rice Krispies in a bowl, and here and there a towering iceberg, sculpted into fantastic shapes, with grottoes and pinnacles of glistening sapphire. The bergs are beautiful but sinister at the same time.

On my final afternoon in Svalbard, rather than embark on another Zodiac expedition, I chose to stay on the ship, and spotted a bear moving across the snowfields at the head of the fjord. In the dull grey light his heavy coat was almost as yellow as a lion’s, and he walked with a slow, rolling swagger as if he owned the entire island.

For an hour I watched him as he made his way around the bay, and even when at last he disappeared from view, his tingling presence still seemed to dominate the horizon.

Brian Jackman travelled as a guest of Exodus Travels (0845 863 9601, www.exodus.co.uk), which offers regular Svalbard voyages from June to August. A 12-day escorted tour starts from £3,217pp, including flights from Heathrow to Longyearbyen (via Oslo), 10 nights in a twin cabin on board the Akademik Sergey Vavilov, a night in a hotel in Longyearbyen or Oslo, and full-board on the ship. A similar trip sets out on July 15 with the wildlife photographer

Jonathan Scott: the Scott in the Arctic 12-day voyage, on board the 48-berth MV Shokalskiy, costs from £3,737pp. Other companies offering Svalbard voyages include Blue Water Holidays (01756 706526, www.cruisingholidays.co.uk) and Discover the World (01737 218800, www.discover-the-world.co.uk)