So we have to simulate the weather

The upshot of this chapter is that in order to forecast global warming, we will have to simulate the time and space variations and imbalances in the energy budget, and the way that the Earth's climate responds to this forcing by storing or transporting heat around. The layer model will not do. Unfortunately, the physics which governs the flow of air and water is complex and difficult to simulate by its very nature. The difficulty is that fluid flow such as in the atmosphere and the ocean takes place on a wide range of size scales. Let's say we want to simulate the flow of the Gulf Stream in the ocean. The Gulf Stream is like a giant river in the ocean. Pieces of the flow spin off into the waters on

Fig. 6.6 A thought experiment in which the bottom of a bathtub is level, that is to say, is on a geopotential surface, but the water level at the surface is sloped. In this situation there would be higher pressure on the geopotential surface at the level bottom of the bathtub on the deeper side than on the shallower side. This results in a pressure force that tends to push water from deeper to shallower.

Fig. 6.6 A thought experiment in which the bottom of a bathtub is level, that is to say, is on a geopotential surface, but the water level at the surface is sloped. In this situation there would be higher pressure on the geopotential surface at the level bottom of the bathtub on the deeper side than on the shallower side. This results in a pressure force that tends to push water from deeper to shallower.

either side. Satellite images of ocean plankton distribution show rings, meanders, wisps, and eddies. The energy of the flow is being converted from the large-scale Gulf Stream into smaller-scale wisps and eddies. In the atmosphere, the large-scale flows break up into storms and weather. Eventually when the eddies get small enough, their energy is absorbed by friction of the flowing fluid. Some fluids like molasses have more friction than others like water; they are said to have higher viscosity. This process of large-scale flow breaking up into smaller and smaller eddies until they get so small that they are absorbed by friction in the fluid is called the turbulent cascade. The physicist Lewis Richardson summed it up in verse

At the heart of it, fluid flow is governed by Newton's laws of motion. Because fluid has mass it has inertia, which tends to keep it moving if it's moving, or keep it stationary if it is already stationary. To change the speed or direction of the flow motion requires a force, such as gravity or pressure. One example of pressure driving fluid flow would be to fill up a bathtub with water and then by magic pull up on the water surface on one side, making a slope (Fig. 6.6), and then let it go. The water will flow to flatten out the surface. The way to see the pressure force is to think about the pressure within the fluid relative to the gravity field of the Earth. When we say "downhill" we mean downhill relative to a surface that is flat, which we call a geopotential surface. The surface of a pool table had better be a good geopotential surface, or the balls won't roll straight. Let's say that the bottom of the bathtub is a geopotential surface. The pressure is higher on the deep side because there is more water overhead. The pressure gradient in the bathtub could be calculated as

Big whorls have little whorls, Which feed on their velocity, And little whorls have lesser whorls, And so on to viscosity.

Pressure gradient

The pressure gradient is high in the bathtub. Our intuition is that the fluid should flatten out. When that happens, the pressure gradient is zero on the geopotential surface.

Stationary observer

Rotating observer

Coriolis fake force

Rotating observer

Fig. 6.7 An illustration of the Coriolis acceleration on a playground merry-go-round. The thrower (T) tosses the ball toward the catcher (C). Before the ball gets to the catcher, the merry-go-round rotates. If we observe this from the point of view of the children on the merry-go-round, it appears as if the ball veers to the right as it moves. The Coriolis acceleration is the kids' attempt to explain the motion of the ball, as if the merry-go-round were not spinning.

The bathtub is behaving the way Newton's laws say it should; a pressure force pushes fluid, and the fluid begins to flow in the direction the force is pushing.

Flows in the atmosphere or ocean differ from those in the bathtub in that the flow in the former persists for long enough to be steered by the rotation of the Earth. Let's ponder this topic sitting on a merry-go-round at a playground (Fig. 6.7). Two kids sit opposite each other on a merry-go-round tossing a ball back and forth while spinning. The thrower (T) tosses the ball, and it goes straight, as Newton would like. Before the ball reaches the catcher (C) on the other side, the merry-go-round has rotated. The catcher misses the ball. What the kids see is that the ball curved, relative to their rotating world on the merry-go-round.

The same situation applies to a weatherman attempting to describe the motion of the winds on the rotating Earth. The way to interpret a weather map is to think of the Earth as if it were fixed relative to the stars. Then add one other magic force, a fudge force called the Coriolis acceleration, which gives you all the same effects as the rotation of the Earth that we are ignoring. The Coriolis acceleration is a fake force that we imagine is applied to any object in motion. Let's return to the ball on the merry-go-round. Our fake force should change the direction of the ball without changing its speed. To do this, the fake force must be at an angle exactly 90° perpendicular to the direction in which the object is moving. In Fig. 6.7, the Coriolis fake force would have to be directed to the right of the direction of motion, so that it would turn the trajectory of the ball to the right. On Earth, the Coriolis force goes to the right in the northern hemisphere, and to the left in the southern hemisphere.

Now we have to jump from the flat merry-go-round to the gravitating sphere. When we rotate the sphere, what does that do to the people trying to read their weather maps? How much rotation do they feel? The best way to envision this is to imagine a Foucault's pendulum. Foucault pendulums are usually set up in an atrium or stairway so that the weight can hang on a wire 10-20 m long. The wire is so long that the pendulum swings with an enchantingly long, slow period of maybe 10 or 20 s. The weight is rather heavy at 100 kg or so. The wire is mounted on a swivel to allow the pendulum to rotate freely if it chooses to. Once the pendulum has started swinging it goes all day. At the base of the pendulum, some museum employee sets up a circle of little dominoes which the pendulum knocks over as it swings. Over the course of the day, the swing direction of the pendulum changes, knocking over a new little block every hour, on the hour, or some other crowd-pleasing rhythm. Leon Foucault installed the first of these into the Pantheon in Paris for the 1855 Paris Exposition.

Now we can use the pendulum to think about the question of rotation on the surface of the Earth. Let's set up a Foucault pendulum on the North Pole (Fig. 6.8). The Earth spins relative to the fixed stars. The pendulum maintains its swing orientation to be stationary relative to the fixed stars. The pendulum knocks over dominoes on both sides of its swing, so they will all be knocked over after the planet has rotated 180°, which will take 12 h.

Next let's move the pendulum to the equator. The trajectories of the pendulums cannot remain completely motionless with respect to the fixed stars because

Fig. 6.8 The Coriolis acceleration that we feel on Earth depends on latitude. A Foucault's pendulum is a rotation detector. A pendulum set in motion at the poles would swing through 180° in 12 h, knocking down all of a circle of dominoes placed around the pendulum. On the equator, the pendulum is started in a north-south direction, and maintains this direction of swing as the Earth rotates. The rate of rotation, and therefore the Coriolis acceleration, is strongest at the poles. At the equator there is no apparent rotation.

Fig. 6.8 The Coriolis acceleration that we feel on Earth depends on latitude. A Foucault's pendulum is a rotation detector. A pendulum set in motion at the poles would swing through 180° in 12 h, knocking down all of a circle of dominoes placed around the pendulum. On the equator, the pendulum is started in a north-south direction, and maintains this direction of swing as the Earth rotates. The rate of rotation, and therefore the Coriolis acceleration, is strongest at the poles. At the equator there is no apparent rotation.

the direction "down" keeps changing as the planet rotates. If we start the pendulum swinging in a north-south direction, the pendulum will keeps its alignment with the stars by continuing to swing north-south. The swing of the pendulum does not appear to rotate to an observer on Earth. The dominoes are safe.

In the middle latitudes, the rate of rotation is in between these extremes of the poles and the equator. The dominoes are completely knocked over in 12 h at the poles, they are never knocked down at the equator. In middle latitudes the knock-down time is longer than 12 h and shorter than forever.

If we release the sloping surface of the water in the bathtub in Fig. 6.6, the water will tend to flow in the direction that the pressure is pushing it, to flow downhill. The driving force for an ocean flow could also be that the wind is blowing (Fig. 6.9). As the fluid begins to flow, the Coriolis acceleration begins to try to deflect the flow to the right (in the northern hemisphere). After a while the fluid is flowing somewhat to the right of the direction that we're pushing it. Eventually, if we wait long enough, the flow will reach a condition called a steady state, a condition in which the flow can persist indefinitely. In the steady state, the Coriolis force balances the driving force, and so there is no net force acting on the fluid. The astonishing implication is that in a rotating world the fluid will eventually end up flowing completely cross-ways to the direction of the forcing! This condition is called geostrophic flow. It is as if the sloping surface in the bathtub would drive a flow across the bathtub. Which direction? Let's stop and figure it out. The pressure from the sloping surface would drive the flow from left to right. An angle of 90° to the right of that would be flowing straight out at the reader, from the page. The water in the bathtub does no such thing, of course, because (i) it doesn't have time to adjust to the steady-state condition in Fig. 6.9, and (ii) the bathtub is not infinitely wide, and water cannot flow through the walls. Great ocean currents that persist for longer than a few days, though, do flow sideways to their driving forces. Sea level on the east side of the Gulf Stream in the North Atlantic is about 1 m higher than it is on the west side. Did I get that right? Figure out the direction that the pressure would be pushing the water, and verify that the flow is 90° to the right.

You can see geostrophic flow on a weather map as cells of high and low pressure with flow going around them (Fig. 6.10). A low-pressure cell in the northern hemisphere has a pressure force pointing inward all around its circumference. At an angle of 90° to the right of that, the winds flow counterclockwise (in the northern hemisphere). Meteorologists call this direction of flow cyclonic. Flow around a high pressure cell is anticyclonic. The fluid flows around the cells rather than into or out of the cells, preserving rather than smoothing out the pressure field that drives it. The ball rolls sideways around the hill rather than directly down the hill as we'd expect. Pressure holes like hurricanes tend to persist in the atmosphere like balls that never run downhill.

Some sets of mathematical equations governing some systems, like the layer model, can be solved directly using algebra to get the exact right answer. The equations of motion for a turbulent fluid are not that easy to solve. We are unable to calculate an exact solution to the equations, but must rather approximate the solution using a computer. A solution we are after consists of temperatures, pressures, flow velocities,

Wind or pressure force

Fluid flow

Coriolis acceleration

Wind or pressure force

Fluid flow

Coriolis acceleration

Wind or pressure force

Flow fluid

Coriolis acceleration

Fig. 6.9 The Coriolis acceleration affects the way that winds and currents respond to pressure forcing on a rotating planet. In (a, when you first turn on the wind) the fluid initially flows in the direction that the wind or pressure force is pushing it. As it starts flowing, it generates a Coriolis force directed 90° to the direction of its motion. In (b, after a few hours) after a while, the Coriolis force swings the fluid flow toward the right. Eventually, the fluid itself flows 90° to the wind or pressure force, and the Coriolis force just balances the wind or pressure force, in (c, the eventual steady state). This is the steady state, where the flow stops changing and remains steady. In the southern hemisphere, the direction of the Coriolis acceleration and the steady-state flow would be the reverse of that shown here.

Fig. 6.10 How pressure variations and the Coriolis acceleration affect weather in the atmosphere. A region of (a) low atmospheric pressure is surrounded by pressure gradients pointing inward, as air tries to flow from high to low pressure (see Fig. 10.6). The steady-state response to that pressure forcing is flow in a direction at 90° to the right of the pressure (see Fig. 9), resulting in counterclockwise flow around the pressure hole. The direction of flow would be the opposite around (b) a high-pressure region, and both would be reversed in the southern hemisphere.

Fig. 6.10 How pressure variations and the Coriolis acceleration affect weather in the atmosphere. A region of (a) low atmospheric pressure is surrounded by pressure gradients pointing inward, as air tries to flow from high to low pressure (see Fig. 10.6). The steady-state response to that pressure forcing is flow in a direction at 90° to the right of the pressure (see Fig. 9), resulting in counterclockwise flow around the pressure hole. The direction of flow would be the opposite around (b) a high-pressure region, and both would be reversed in the southern hemisphere.

and other quantities related to the flow or the drivers of the flow. These quantities vary from place to place, and as a function of time. To represent them in a computer model, the atmosphere and the ocean are diced up into a three-dimensional grid of numbers (Fig. 6.11). There is a temperature, a pressure, velocities, and other variables associated with each grid point. The computer model calculates the pressure force at each grid point by comparing the pressures at adjacent grid cells. The model steps forward in time to predict the response of the fluid velocities to the pressure forces. A typical time step for a climate model may be a few minutes. The flows carry around fluid, altering the pressure field, and the simulation snakes it way forward through time. If you would like to play with a climate computer model, you can download one called EdGCM (see Further reading). The model runs on desktop computers and can simulate a few years of simulated time in 24 h of computer time. It has the facility built in to plot maps and time series, such as temperature as a function of time.

Weather simulation has always been a problem ideally suited for computers. Even before computers, researchers like Lewis Richardson dreamed of using human computers to perform the immense numbers of calculations that would be required. Doing the math by hand is so slow, however, that it would never be possible to even keep up with the real weather, let alone make a forecast of the future. ENIAC, one of the first electronic computers constructed in 1945, was used for a range of scientific computation projects including weather forecasting. The fastest computer in the world as of 2005 is called the Earth Simulator, which is in Japan.

Simulation of weather and climate remains one of the grand challenges in the computational sciences. This is because the mechanisms that govern fluid flow often operate at a fairly small spatial scale. If we wanted a model to include everything that governs how cloud drops form, we would have to have grid points every few meters in clouds for example. Fronts in the ocean can be only a few meters wide. To get the answer right, we'd like to have lots of grid points. The trouble is that if we increase the number of grid points by, say, a factor of 10, in each dimension, the total number of grid points goes up by a factor of 10 x 10 x 10 or 1000. That's 1000 times more mathematical operations the computer has to perform in order to do a time step. To make matters worse, as the grid gets more closely spaced, the time step has to get shorter. A grid 10 times finer would require a time step about 10 times shorter. So it would take 10,000 times longer to do a model year of time. State-of-the-art computer models of climate are run at higher resolution all the time, and in general they look more realistic as resolution increases. But they are still far from the resolution they would like, and will be so for the conceivable future.

Some of the physics of the real atmosphere cannot be explicitly resolved, so they must be accounted for by some clever shortcut. The real formation of cloud droplets depends on small-scale turbulent velocities, particle size spectra, and other information which the model is unable to simulate. So the model is programmed to use whatever information it does have, ideally in some intelligent way that might be able to capture some of the behavior of the real world. Cloud formation may be assumed to be some simple function of the humidity of the air, for example, even though we know reality is not so simple. The code word for this approach is parameterization. The humidity of the air is treated as a parameter that controls cloudiness. Other important and

OS OS

Fig. 6.11 Surface wind field from a state-of-the-art climate model (a model called FOAM, courtesy of Rob Jacob). This is a snapshot of the winds averaged over one day in the middle of January. The figure is lightly shaded to show sea level pressure, with darker shading indicating lower pressure. One can see the wind blowing counterclockwise around low pressure areas in the northern hemisphere, for example, just to the south of Greenland. In the southern hemisphere, winds blow clockwise around low-pressure areas, for example, in the Southern Ocean south of Australia.

ffi o c

Fig. 6.11 Surface wind field from a state-of-the-art climate model (a model called FOAM, courtesy of Rob Jacob). This is a snapshot of the winds averaged over one day in the middle of January. The figure is lightly shaded to show sea level pressure, with darker shading indicating lower pressure. One can see the wind blowing counterclockwise around low pressure areas in the northern hemisphere, for example, just to the south of Greenland. In the southern hemisphere, winds blow clockwise around low-pressure areas, for example, in the Southern Ocean south of Australia.

potentially weak-link parameterizations include the effects of turbulent mixing, air/sea processes such as heat transfer; the list goes on. We will discuss these in more detail in Chapter 11.

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Solar Panel Basics

Solar Panel Basics

Global warming is a huge problem which will significantly affect every country in the world. Many people all over the world are trying to do whatever they can to help combat the effects of global warming. One of the ways that people can fight global warming is to reduce their dependence on non-renewable energy sources like oil and petroleum based products.

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