Disclaimer and beta thanks in Part 1.

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Chapter 8
Friday, November 18, 2005
3:35 P.M.
Charlie's office

Charlie frowned at the chalkboard in concentration, the combination of Larry and Alan's voices in the background a dull buzz in his ears. He tentatively wrote an expression on the board, erased part of it, and then, more confidently, rewrote it and added a final constant at the end.

"Okay, here it is," he said, turning around to face the other two men. Alan was seated in Charlie's desk chair, scrutinizing the blackboard. Larry was perched on the desk, turning a balsa-wood tetrahedron over and over in his hands, and saying something about lost opportunities and the small tragedies of life.

Laurel again, Charlie thought. His friend's relationship, after veering off the trail of friendship last year, seemed to have continued right on into the ditch. Larry hadn't talked to Charlie very much about her, but Amita told him that Larry had quietly thanked her for her advice to try again before telling her some things apparently weren't meant to be. He remembered Amita looking down at the floor as she said it, just a week or so ago, and wondering if it was really Larry and Laurel she was talking about. He dragged his thoughts away before they could go any farther down that well-worn trail of regret.

Alan and Larry were both looking at him expectantly, and he cleared his throat. "So, um, Dad, these are the equations you asked me to put together. Basically, it's an isothermal isotropic one-dimensional chemical transport equation, modified for the unsaturated ground conditions of our particular site."

Alan blinked. "No wonder I was having a hard time figuring it out myself," he said dryly.

"Actually, you were on the right track." He was several miles away from the station, to continue the railroad metaphor, but Charlie wasn't about to tell him that. "This is just the first step, though. See, what we're interested in modeling is something like this…"

Turning back to the board, he drew a series of evenly-spaced dots in a grid pattern, then drew lines between them to make them look three-dimensional. About halfway down, he shaded in the spaces between some of the points. Then he added a heavy line across the top of his lattice, and drew a child's version of a tree on top just for fun. "Think of a network of points, or nodes, underground. You might think of groundwater as being like an underground lake, but really it's found in-between particles of soil, rock, and whatever else is down there. Above a certain point–" he tapped the top part of the lattice drawing–"there's more air than water between the soil particles. Below that," he said as he indicated the lower portion of his diagram, "the spaces between the soil particles are saturated with water."

"Yes, Charlie, I know that. And that line between them is the water table," Alan said.

"Right. Now, say we inject something else besides water into the soil." He drew a little rocket next to his tree. As he turned back to his audience, he saw his father smirking at him. "What? The most common source of perchlorate is solid rocket fuel."

Alan shook his head. "You're going to be great with your kids someday, you know that? I can just see you explaining higher mathematics to a three-year-old using little pictures like that."

Charlie flushed. "Hey, my students seem to like it." He rushed on to deter any further mention of future grandchildren. "Anyway, the point is, if something like perchlorate leaks into the ground—" he took a piece of yellow chalk and drew a line coming out of the rocket and into the lattice—"it's going to move differently in the unsaturated zone, where there's still air between the soil particles, than below the water table, where it's interacting with water."

He took his chalk and tapped one of the dots in the lattice. "So what we do is draw an imaginary network of points, or nodes, underground. We run this equation, which is for one-dimensional flows only, between each of these points. When we put the results together, we get the three-dimensional conclusion."

Alan nodded. "And what do you get?"

Charlie nodded to the desk behind him. "See that big stack of paper there? After several hours of computing time, that's what you get."

Alan laid a hand on top of the pile, which was about four inches thick. "Is there a short version?"

Larry chuckled, and Charlie said, "Of course. I plugged the results into some 3-D visualization software, and here's what I got." He reached between Alan and Larry to wake up his laptop and click on the appropriate window. What they saw was a reproduction of the lattice he had drawn on the board, but with different colors assigned to the different points.

Alan regarded it for a moment. Then he said, "If I'm interpreting this right, and please tell me if I'm not, it looks like the plume of perchlorate doesn't get very far." He pointed at the screen, where a cluster of red dots trailed away into orange and yellow, with a few green dots at the bottom edge of the screen.

"That's right. Which means that this is accurately modeling what we already know. The city only closed a couple of wells, remember? Which means that they, and NASA, knew that this was a limited release. Based on the data you gave me, Dad, I've verified that."

"Your clients should be pleased, Alan. If they were seeking to determine the threat to the Los Angeles River posed by perchlorate, this is felicitous news."

Alan looked thoughtful for a moment. "Yes, but..." He turned to Charlie. "Two things. First, if it turned out that this was a relatively harmless spill, they also wanted to know how large a release it would take for the worst to happen. Can you run your model backward? In other words," he pointed to the edge of the screen, "can you start with the assumption that the perchlorate makes it all the way through the aquifer, and work backwards to figure out how much would have had to be released?"

"Sure." Charlie thought for a moment. "As long as you tell me what a harmful concentration would be at the endpoint, I can track that back through all the nodes."

Alan snorted. "Well, that's a loaded question. And that's my second point."

Charlie cocked his head to the side. "What do you mean?"

His father leaned forward, forearms on his elbows. "See, perchlorate has only been a problem for the past seven or eight years. Before then, we didn't have the technology to test for it like we do now. And there just haven't been enough studies done on how harmful it is."

"What does it do, anyway?" He hadn't been thinking about the health ramifications of the equations and models he'd been working with, just the geometry and fluid dynamics of different substances moving underground. He felt like he was coming up to the surface after scuba diving, realizing that the world below him, while completely fascinating, was qualitatively different than the world he was now in. It took him a second to readjust his thinking.

"It's not as bad as far as contaminants go," Larry was saying, examining the tetrahedron he still held. "It mimics iodine and inhibits the synthesis of growth hormones in the thyroid gland."

"That sounds pretty bad," Charlie retorted.

Alan shook his head. "For most people, it doesn't do anything; we're full-grown and we get enough iodine from our daily diets. It's for kids, and especially pregnant women, that it's a concern."

"What's really interesting," Larry interjected, "is that perchlorate is a naturally-occurring substance. It's found in drinking water supplies in Texas and Chile, among other places, and studies there show no ill effects whatsoever to children or fetuses. Now, man-made perchlorate can be chemically distinguished from the natural variety, and so it's possible that the artificial substance can do harm which the natural one can not."

"In other words, it's not as bad a lot of the stuff that could be seeping out of JPL," Alan concluded. "Over fifty potentially toxic chemicals, according to the most recent information I have. This has just gotten a lot of press because it's relatively new, and there's a lot still unknown about it. So anyway, the short answer to your question is, the state of California says six parts per billion is the safe limit."

Charlie pictured an Olympic-size swimming pool full of water, with six small drops of perchlorate floating somewhere among the other hundreds of millions of drops. "That's a pretty high standard."

"Better safe than sorry, right?" Alan shook his head. "Besides, my point is that there's much more harmful chemicals that these guys could be worried about getting into the river, and into the groundwater. I just wonder why they picked this stuff."

He shrugged and stepped back. "Like you said, it's gotten a lot of press. Maybe when you tell them that it's not really that harmful, they'll look in some other direction."

"Maybe so. Anyway, it's six parts per billion, Charlie. Let me know how long it takes you to figure out that calculation, and I'll add it to my bill."

He waved his hand. "No, Dad, it's okay."

"It's not like I'm paying you, my client is. And you're putting a lot of work into this, so you should be compensated for it."

"Really, you don't have to. Look, I've been trying to think of a new example for my applied mathematical modeling class, and this is perfect. It would have taken me the same amount of time to prepare something for class, so if I were billing my hours, that's what I'd call it: class prep."

Alan looked at him for a moment, then nodded his assent. "All right, have it your way. I appreciate it, Charlie."

"Don't mention it." He waved his hand again.

Larry followed Alan out of the room, waving at him as he left. Charlie raised a hand in response and stood there thinking for a moment. He was glad his father hadn't pressed him on the issue, because he was reluctant to reveal the reason behind his refusal of payment. It wasn't that he didn't need the money, which he didn't. It wasn't that he would try to use the material in class, which he would; it was a good, solid example that was even locally relevant.

It was something that he'd felt with Don a few times now. His family had spent so many years on him, making sure his unusual genius was properly nurtured and applied. They would rarely, if ever, admit to his face that it had been difficult, but he knew it had, emotionally for Don, and financially as well as emotionally for his parents. Here was his chance to take that potential they had enabled him to achieve, and put it to use for them. He enjoyed not just helping the FBI, but helping his big brother with the part of his life that had been closed off to him ever since he joined the Bureau.

And right now, he was enjoying the opportunity to be of more use to his father than planning out a seating chart for a wedding.

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