DIVERGENCE SYNDROMES
I spend a lot of time walking on the beach, thinking about the past and my life so far. So far, so good? Well, you will be able to make up your own mind, and so, perhaps, will I. My work is easy and I finish it quickly. I have plenty of time to remember.
Fragments of John Clearwater’s conversation come back to me. When he was working on turbulence, he told me he had had such good results because be had decided to tackle the subject in a new way. In the past, he said, people tried to understand turbulence by writing endless and ever more complicated differential equations for the flow of fluids. As the equations became more involved and detailed, so their connection to the basic phenomenon grew more tenuous. John said that his approach was all to do with shapes. He decided to look at the shapes of turbulence and, immediately, he began to understand it.
It was at this time that his talk was full of concepts he referred to as Divergence Syndromes. He explained them to me as forms of erratic behavior. And in a subject like turbulence, naturally, there will almost always be a divergence syndrome somewhere. Something you expect to be positive will turn out to be negative. Something you assume will be constant becomes finite. Something you take confidently for granted suddenly vanishes. These are divergence syndromes.
This sort of erratic behavior terrifies mathematicians, John said, especially those of the old school. But people were learning, now, that the key response to a divergence syndrome was not to be startled, or confounded, but to attempt to explain it through a new method of thought. Then, often, what seemed at first shocking, or bizarre, can become quite acceptable.
As I stroll the length of this beach I consider all the divergence syndromes in my life and wonder where and when I should have initiated new methods of thought. The process works admirably with benefit of hindsight, but I suspect it wouldn’t be quite so easy to apply at a moment of crisis.
It was at Sangui, João’s village, that the tarred macadam road began. I turned onto it, heard the empty trailer, towed behind the Land-Rover, bump up over the curb and settled down for the long drive into town. Normally it took between four and five hours, but that was assuming there were no major accidents on the way, that the bridges were in reasonable repair, that there were no protracted delays at the numerous military roadblocks and that you didn’t get caught behind one of the supply columns returning from provisioning the federal troops fighting in the northern provinces.
I rather enjoyed this drive—I had done it three times before—and on each occasion relished the buoyant end-of-term sensations it provoked. Turning off the laterite track in Sangui onto the crumbling, potholed tarmac of the main road south was like crossing a border, a frontier between two states of mind. Grosso Arvore was behind me, I was on my own for a few days. Almost alone: two kitchen porters, Martim and Vemba, sat in the back of the Land-Rover on piles of empty sacks. I had offered them the front seats, as I always did, but they preferred their own company in the rear.
The road was straight, running through dry scrubland and patchy forest that spread south from the hills of the escarpment behind me to the ocean, two hundred miles away. It was early morning and the sun was just beginning to burn off the dawn haze. The routine was familiar. The first day was occupied getting to the town. I would spend the night at the Airport Hotel and the next day would be made up of an enervating round of visits to the bank and department store and the various merchants who provided the project with food and supplies, black market drugs and medicines. Occasionally, there were trips to be made to workshops and garages for machinery to be fixed, or spare parts searched for, and this could add an extra day or two to the trip. But on this occasion I was merely provisioning. A long day’s shopping awaited me tomorrow. Then I would spend one further night at the hotel before heading back for home, a much slower undertaking, with the Land-Rover and its trailer heavily loaded. Thirty miles an hour was our average speed.
The road ran through an unchanging landscape. Every ten miles or so we would encounter a small village. A cluster of mud huts thatched with palm fronds; a few traders’ stalls set out on the verge selling oranges and eggplant, sweetmeats and cola nuts. The journey was not dangerous—the fighting was distant and only the federal army had aircraft—but we were always warned not to attempt it after dark. Ian Vail had broken down once, and was very late returning, but Mallabar had refused to send out a search party for him until the next morning. I was never absolutely clear what we were meant to be frightened of. Brigands and bandits, I supposed: there was a risk of highway robbery after dark. Apparently there were gangs roaming the countryside, composed mainly of deserters from the federal army. It was these men that the many roadblocks were designed to deter or catch. Every half hour or so one would come across these outposts, nothing more than a plank of wood propped against an oil drum jutting out into the road, and beyond it in the fringe of the bush or beneath the shade of a tree, a lean-to or palm frond shelter containing four or five very bored young soldiers wearing odd scraps of uniform. You had to slow down and halt whenever you saw one of these oil drums. Someone would peer at you and then, usually, motion you onward with a lethargic wave. If they were feeling bloody-minded they would make you step out of your vehicle, examine your papers and make a cursory search.
These were the moments I did not enjoy particularly: standing in the sun beside the Land-Rover being scrutinized by a young man in a torn undershirt, camouflage trousers and baseball boots, with an ex-Warsaw Pact AK47 slung over his shoulder. It always seemed especially quiet at that moment. It made me want to shift my feet, or cough, just to break the silence that pressed around me as the soldier examined my laisser-passer. In the half dozen times I had been stopped, never once had another car or lorry driven by. It was as if the road belonged exclusively to me.
On this journey, though, we were being waved through without exception. The mood of the men seemed more jocular, and more than once as I had driven off I had seen beer bottles being raised to lips. I remembered what Alda had told me about the defeat of UNAMO forces. Perhaps this was a prearmistice relaxation and the war would be over soon.
We reached the Cabule River by late afternoon. The ramshackle buildings on the far bank marked the outskirts of the town. Our wheels rattled noisily on the metal planking of the ancient iron bridge. The river was four hundred yards wide here. It took a great slow swerve around the town before disgorging its brown water into the dank creeks of its mangrove-clogged delta ten miles away down the coast. The edge of the continent ran straight here—mile after mile of beach and thundering surf. The silty Cabule was navigable only by vessels of the shallowest draft. All the bauxite from the mines—this province’s major source of wealth—had to be transported to the capital and its harbor by rail. Bauxite mines, some timber, a few sugar and rubber plantations, sharecropping and the Grosso Arvore National Park were all this area of the country had to recommend it.
I drove slowly through the town. On either side of the road were deep ditches. A few brick buildings housed empty shops and drinking dens. In the mud-walled compounds beyond them smoke rose from charcoal fires as the evening meal was prepared. The first neon lights—ultramarine and peppermint—flickered in the shack-bars and on the concrete terraces of the hotel-brothels and nightclubs. Music bellowed from loudspeakers perched on roofs or hung from rafters. In the crawling traffic, taxi drivers sat with their fists pressed on their horns. Children knocked on the side of the Land-Rover trying to sell me Russian watches, feather dusters, yo-yos, felt-tip pens, pineapples and tomatoes. There were many soldiers on the streets, carrying their weapons as unconcernedly as newspapers. Old men sat on benches beneath the dusty shade trees and watched naked children spin hoops and chase each other in and out of the rubbish bins. At an uneven table two young spivs with shiny shirts played stylish Ping-Pong, stamping their feet in the dust and uttering hoarse cries of bravado as they ruthlessly smashed and countersmashed.
The press of traffic nudged its way through the town cent
er, past the five-story department store and the mosaic-walled national bank with its swooping modernist roof; past the white cathedral and the brutalist Department of Mines; past the police station and the police barracks, with its flagpole and ornamental cannons, the neat stacked pyramids of cannonballs like the swart droppings of some giant rodent.
Then we turned and headed back north again on the new road to the airport, past the hospital and the exclusive, fenced-in suburbs. We drove past the convent school—St. Encarnación—past the shoe factory and the motor parks. The setting sun basted everything with a gentle peachy light.
The airport was far too large for such an undistinguished provincial capital. Built shortly after independence in 1964 by the West German company that owned and ran the bauxite mines, it was designed to take the largest commercial jets (optimism is free, after all). A sprawling modern hotel was constructed nearby to accommodate all the projected passengers. The bauxite was still being extracted, the mines and the processing plants functioned, after a fashion, but the airport and its white hotel were always heading for decline and desuetude. Five arrivals and departures a day were all it boasted, domestic flights linking other provincial cities. Air Zambia flew in once a week from Lusaka, but the much heralded UTA link to Brazzaville and Paris became another casualty of the civil war when rumors spread that FIDE, or was it EMLA?, had been sold ground-to-air missiles by the North Koreans.
The war had benefited the airport in other respects, however. Half the federal government’s air force was based there now: a near squadron of Mig 15 “Fagot” fighters, three ex-RAF Canberra bombers, half a dozen Aermacchi trainers converted to ground attack and assorted helicopters. As we drove past the perimeter fence I could see the old Fokker Friendship revving up at the end of the runway about to depart on its evening flight to the capital, and beyond it, in their bays, the tubby, tilted-back silhouettes of the Migs.
At the hotel I said good night to Martim and Vemba, agreed to the time of our rendezvous the next morning, and checked in. The hotel was distinctly shabby these days, all incentive to keep it spruced up having long gone, but, after weeks at Grosso Arvore and my tent, it seemed to me still redolent of a tawdry but alluring glamour. It had a restaurant, a cocktail bar and a half olympic-sized swimming pool with a barbecue area. Its rooms were contained in two-story annexes, connected to the main building by roofed-over walkways that passed through tropical gardens. Scattered here and there were one- and two-bedroom bungalows for those guests who planned a longer stay. Sometimes, piped Latin-American music was played in the lobby. The staff wore white, high-collared jackets with gold buttons. At the entrance to the restaurant a notice requested, in English: LADIES PLEASE NO SHORTS. GENTLEMEN PLEASE TIES. Whether it was the ghosts from the heady days of the bauxite factory contractors’ ball, or the still lingering pretensions of the current management, the Airport Hotel (this was its evocative name) had an ambience all its own. It also had air conditioning, sometimes, and hot and cold running water, sometimes, both luxuries that were permanently absent at Grosso Arvore.
I walked through the unkempt gardens to my room, unpacked, had a shower and changed into a dress. I felt fresh, cool and hungry.
I strolled along a walkway to the main building. It was now quite dark and the warmth of the night air, after the chill of my room, seemed to lie gently on my clean bare arms and shoulders like a muslin shawl. I could hear some rumba Muzak wafting over from the lobby’s sound system and from all around me in the grass and bushes came the endless creek-creek of the crickets. I stopped and filled my lungs, smelling Africa—smelling dust, woodsmoke, perfume from a flower, something musty, something decaying.
I turned onto another path and quickened my pace toward one of the cottages. Its windows were shuttered but I could see light shining behind them.
I knocked on the door and waited. I knocked again and it was opened.
Usman Shoukry looked at me, not surprised, but trying not to smile. He was wearing loose linen shorts and a lilac T-shirt. His hair was shorter than the time I had last seen him.
“Look who’s here,” I said.
“Hope,” he said, deliberately, as if he were christening me. “Come on in.”
I did, and he shut the door. When I kissed him I stuck my tongue in his mouth and slipped my hands under his T-shirt and felt his back, running them up to his shoulder blades and then down beneath the waistband of his shorts, my palms resting lightly on his cool, hairless buttocks.
I broke away from the kiss, still holding him to me. His mouth was glossy with saliva. He rubbed it dry with the back of his hand, smiling at me again. I looked at him as if I hadn’t seen him for years. The sherry color of his brown eyes, his slightly askew nose, his thick lips.
“Are you in trouble?” he asked.
“No. Why?”
“I wasn’t expecting you for two months.”
“Well, it’s your lucky day, then, isn’t it? Come on, let’s eat before they run out of food.”
THE INVERSE CASCADE
Hope Clearwater buys four parsnips in the market today. She is delighted and very surprised to find them. She asks the trader where they came from. Nigeria, the trader says. Hope doesn’t believe her, and questions her skeptically: “Where exactly in Nigeria?” The trader is not pleased to have her word doubted. “Jos,” she says, and turns away. Hope remembers that Jos is situated high on a plateau in central Nigeria. All sorts of fruit and vegetables can be grown there because of its cool nights and dry days—even raspberries and strawberries.
The parsnips remind her of a story John told her about an old professor of his. This man had worked for a time on the problem of turbulent diffusion. For his experiments he required a large number of floats that had to be at once highly visible and not affected by wind. For this reason balls—both rubber and Ping-Pong—would be of no use. After experimenting with turnips and potatoes, the professor discovered that the ideal vegetable was the parsnip. Its rough, conic configuration, and the fact that in the water most of the vegetable was below the surface, made it a stable float and indifferent to all but the severest breezes.
What the professor did was to paint several dozen parsnips white and tip sackfuls of them off a bridge over the River Cam. The white parsnips would float downstream, be caught in eddies, cluster and circle in side-swirls or flow downriver in long bobbing strings, photographed by the professor’s assistants standing with cameras on both banks of the river at intervals of twenty yards.
The professor (Hope can’t remember his name) bad done useful work, John said. The problem was that, despite his imaginative experimental technique, his thinking was too rigid. He believed that turbulence was caused by a cascade of energy from large eddies to small. But John’s own work—his breakthrough, as he termed it—had shown that this was only part of the story. In every case of turbulence, in whatever medium, there is also an inverse cascade, a flow of energy from the small eddy back to the large. Hope remembers clearly the day John proved this. He had explained it to her, his voice hoarse with excitement. Disorder, he said, is not simply handed down a chain; some of it is always being handed back again. Once that fact is grasped, a great deal that was baffling about turbulent systems becomes far easier to understand.
When John Clearwater came back from America he was in good form. He had met someone at the conference—a statistician—who had helped him enormously, almost without knowing it. He told Hope about these new avenues that had opened up, new potentials he could now see. Hope laughed with him, with genuine pleasure—with relief perhaps—at this excitement. Once again what he told her—he spent two hours trying to explain it to her—meant little or nothing, but she felt pleased, reassured. That time, after his return from America, calmed her, staunched the thin hemorrhage of anxiety and doubt. All, it seemed, was well again.
He worked as hard as ever, leaving the flat at eight in the morning and not returning, usually, until nine at night, but their hours together seemed to Hope to recapture some of the
vivacity and edge of their first months of marriage. Later, when she looked back, she realized that they had merely been going through another phase. (She found she could demarcate the phases in her married life as efficiently as a historian—they seemed as precise as the circles of growth in the trunk of a tree.) In this particular phase a new enthusiasm dominated—the cinema. They went fairly frequently to the cinema and theater, whenever the mood took them, or whenever some triumphant succés d’estime seemed to demand it. But now John wanted to go out every second or third night. And at first it was fun. His absorption was so intense and single-minded, and the pleasure he took in the cinema so manifestly good for his spirits, that it was a privilege, she reckoned, to share in it. But after six weeks of this, amounting to over two dozen visits—some films were returned to two or three times—she began to find that the strain of accompanying him was growing and she started making excuses.
Part of the problem was that he insisted sitting very close to the screen, in the front row preferably, and certainly no farther back than the third, so that his entire field of vision was dominated by the projected image. At first this was oddly exhilarating, and Hope would emerge from the cinema with her head reverberating like a gong from the big booming pictures, breathless and jangled.
His other idiosyncrasy, however, was harder to take and began to irritate her. He was scrupulous about the type of film they saw. Reviews in as many newspapers as possible were studied and collated, and he built up a small library of film reference books in an attempt to ensure that the film they were going to see would fulfill the demands he made of it. She accused him, jokingly at first, of being the only person she had ever met who sought to entertain himself in a wholly prescriptive way. It was the very opposite of random; he wanted to take no risks. “How can you enjoy yourself, how can you have fun, without an element of risk?” He paid no attention. It was not a narrow censorship he indulged in—he was very keen on horror films and violent thrillers—it was simply that he believed, with a fundamentalist’s zeal, that a true film, a film that was true to the nature of its own form, had to have a happy ending.