*stretch*
All finished with my physics paper.
*phew*
Now i've only an exam (french), and three finals left.
Hopefully the weather will warm up.
And hopefully I'll secure a lab job before summer's end.
There's a chance I might sign up for a sublet today, too.
If you feel like reading a boring paper, or seeing what I was up to...
---
In Madison, Wisconsin, if you drive out of the city a few miles on a cold winter’s night and look towards the northern sky, you might be blessed with the spectacular sight of the shimmering red and green flames of the Aurora Borealis. Billions of electrons zoom along the Earth’s magnetic field lines to the poles, violently colliding with gas atoms in the Ionosphere, and releasing light in an awe-inspiring display the entire northern hemisphere can observe . Similarly, if you drive a few miles out of the city and go down underneath the expansive cornfields, you might find the underground bunker used by the Madison Dynamo Experiment to understand the very same magnetic field that gives you the Northern Lights .
It is a well-known fact that the Earth, Sun, and other astrophysical bodies have strong magnetic fields. What is not understood is how these fields developed, nor why they behave the way they do. Professor Cary Forest leads the investigative research project at the University of Wisconsin-Madison in what is called the Madison Dynamo Experiment (MDE). Together with his team of researchers, they hope to answer the many important questions concerning dynamos that remain unsolved: How fast do naturally occurring magnetic fields grow? When do they stop growing? What causes them to stop growing? What types of flow are required for a laminar dynamo (a phenomenon dealing with magnetic fields in astrophysical bodies), and what are the effects of turbulence in a dynamo? These questions are answered in the three major stages of the MDE, that is, the theory, the water vessel experiment, and the sodium vessel experiment.
The primary cause of the strong magnetic fields in astrophysical bodies is generally accepted to be the “dynamo effect.” To begin, a young or forming celestial body (such as a planet or star) has a weak magnetic field. Its interior is made up of various flowing materials or plasmas; in the case of the Earth, this would be molten iron. The flowing of these conducting fluids causes the magnetic field lines to bend and, more importantly, stretch, thereby amplifying the total magnetic field. If the resulting magnetic field continues to grow, eventually it will affect and control the flow of the conducting fluids. Without the need for the original magnetic field, you now have a self-sustained dynamo.
The theory behind the growth and behavior of the magnetic dynamo is described by the appropriately named ‘dynamo equation’, as well as a combination of the magnetic induction equation and the Navier-Stokes equation. The dynamo equation is derived directly from three fundamental laws (Ampère’s Law, Ohm’s Law, and Faraday’s Law), and is essentially a watered down form of the Navier-Stokes equation, with a term included to represent the Lorenz force. The Navier-Stokes equations are partial differential equations that describe the flow and velocity of incompressible fluids. The development of a solution to these equations applicable to astrophysical dynamos was one of the most difficult (and important) obstacles in the theory leading up to UW-Madison dynamo research.
To solve this, the assumption was made that the original magnetic field was weak (and thus the Lorenz force), thereby decoupling the two equations. Consequentially, the linear partial differential equation could be solved.
Two vital discoveries were made from this accomplishment. The first addresses the problem of magnetic field diffusion. Although it may seem obvious, it’s drastically important that the flows in the fluid stretch and bend the magnetic field quicker than they can deteriorate away. As the ratio of energy transfer to diffusion is fundamental to understanding and improving dynamos, it is given a specific name: the magnetic Reynolds number, Rm. The value of this number is directly dependant on the material’s viscosity, conductivity, and size. As the Reynolds number increases for a fluid, the growth rate of the magnetic field grows larger. Since growth is a rate of change, it has a critical number that occurs when it passes from negative to positive values; this number is known as the critical Reynolds number, and determines when the system becomes a dynamo. Any fluid with a Reynolds number below this critical value will have a magnetic field that decays over time, while the magnetic field in a fluid with a Reynolds number greater than the critical value will grow larger.
Obviously, in studying dynamos one of the goals is to reduce the critical Reynolds number in the fluid by as much as possible. This highlights the second discovery made by the MDE: that optimizing the critical Reynolds number can be achieved by adjusting the geometry of the flow to create positive magnetic feedback. The geometry of the flow being studied in the MDE is referred to as the ‘stretch-twist-fold’ dynamo. As the name sounds, the magnetic field lines in the fluid are stretched, twisted up and around, and then folded over like kneaded bread dough. The result is a tangled mess of field lines, plus a new field line situated in the same space and orientation as the original, indicating a new magnetic flux which can then be altered all over again.
Although in the MDE the fluid mechanics of the dynamo seemed to be understood on paper, experimental data to base the final project’s blueprints off of were needed, and as a result the water version of the MDE was built. Under the high velocity (roughly 20 m/s) and high pressure conditions of the spherical dynamo used by Madison researchers, water (due to its viscosity and mass density) behaves almost identically to the conducting fluids planned for the magnetic experiments. Considering its availability and low cost, water was the perfect substitution for studying the motion of particles inside a dynamo and the optimization of apparatus specifications.
In large physical bodies, heat transfers similar to convection pockets provide the kinetic energy needed for a self-sustained dynamo. In the lab setting, propellers are used to imitate this behavior. For the water-based experiments, a stainless steel spherical shell with a diameter of one meter was used . Inside the sphere were two impellors, evenly spaced apart, that rotated in opposite directions. It was fitted with five viewing windows for the purpose of measuring flows and observation. All of the calculations and measurements were done in a spherical coordinate system (as it is the most natural in this setting), and consequently the windows provided a full scan of the apparatus for both r and θ dimensions. A process called Laser Doppler Velocimetry was used to take the actually measurements and collect data on flow geometries. To prevent cavitation at high flow speeds, the windows were also designed to withstand pressures up to 670 kPa.
Unfortunately, a problem arises from using propellers as a substitute for massive astrophysical convection processes. Because higher velocities must be attained, turbulent nonlinearities arise in the flow. In some cases (known as the beta effect), cyclic turbulent patterns in the laminar dynamo can transport magnetic flux out of the system and increase diffusivity, much like the stirring at the edge of a river reduces the net momentum of the flow. The fluid’s conductivity is also reduced, and thus the critical Reynolds number is increased, making it more difficult to produce the dynamo. Whether or not this problem would significantly affect the magnetic-based portion of the research was a major issue for the early stages of the MDE.
Luckily, there was a second, inverse form of turbulence observed that possibly eliminates worry over diffusion due to the beta effect. Tiny helical motions in the flow mirror the system’s behavior and can produce the same stretch-twist-fold on a much smaller scale. These helicies produce current, and thus magnetic field lines. If the flow is carefully tweaked, this form of turbulence (aptly called the alpha effect) can create a positive net magnetic field, and can in fact greatly contribute to forming the desired dynamo.
Water was used primarily to see if the desired flows could be created in the lab using propellers. Fortunately, it was shown that the propellers could be designed specifically to create the desired eigenmode (that is, the least damped, fastest moving eigenmode) and thus the desired dynamo.
The next step was to build the vessel to be used in producing dynamos and studying their behavior. Construction of this apparatus finished just recently with the insertion of the impellors into the vessel. In contrast with the water experiment, the stainless steel sphere has a diameter of 0.5 m, and operates on 150 kW of power (roughly divided between two 75 kW motors for the impellors). It can handle a maximum of 100 kPa of pressure, and peak mean speeds of up to 20 m/s. Instead of windows, eight 5 cm ports were built around the sphere for the insertion of probes.
Several assumptions had to be made in the construction of the second vessel and in making predictions of its results. The first deals with the material that is to be used as the fluid in the dynamo, that is, sodium. If the results from the water experiment are to be extrapolated to the sodium experiment validly, it must be assumed that the flows in the sodium will be identical to those observed in water. This assumption is safe because of the extensive similarities between water and sodium’s physical properties. Both have nearly the same mass density. Also, the viscosity of sodium at 120°C is approximately identical to that of water at 40°C-60°C. Next, it is assumed that the mean velocity field’s shape will not change as power is increased. This is important due to the fact that a larger amount of power will be required for the sodium experiment than was needed for the water experiment. Finally, the other major assumption is that in the UW-Madison lab setting, turbulence is negligible and thus can be ignored. Of course, this is a major assumption, and is justified by the fact that the measured mean quantities are significantly larger then those of the turbulence, and by the lack of a strong Lorenz force inside the lab-based dynamos. Still, it is the one soft spot of the research, and will undoubtedly move to the spotlight as our general understanding of dynamo behavior grows.
With this setup, the flow is expected to achieve Reynolds numbers greater than 100 , which should be significantly higher than the critical Reynolds number for the sodium conducting fluid. Therefore, the system’s magnetic field is expected to grow, until eventually the Lorenz force does become strong and controls the flow geometry of the system. At this point, the conductivity of the fluid is expected to decrease, thus raising the critical Reynolds number. Although this may sound undesirable, it is predicted that when the critical Reynolds number and Reynolds number from the water dynamo experiment intersect, they will flat-line, and the system will become a saturated self-sustained dynamo (a result from the complete solution to the Novier-Stokes/induction equation including the Lorenz force).
In truth, the Madison Dynamo Experiment is just entering into the most exciting phase, where predictions are checked and hopes are met or lost. Now that the sodium dynamo vessel has been built, regular testing and refinement can proceed. It is expected that this phase of the experiment will span close to a decade. Major areas of interest include lowering the critical Reynolds number, understanding saturation more thoroughly, accurately predicting the growth of the dynamo, and refining dynamo theory to account for turbulence. The most important (and interesting) of these is the inclusion of turbulence, as it makes the system significantly more complex and unpredictable. It is very likely that after the beginning sodium tests are complete, the MDE will focus more intensely on furthering our knowledge of turbulent dynamo systems.
In the end, the Madison Dynamo Experiment is a perfect example of research done for the sake of scientific curiosity, and the urge to gain a better understanding of the world around us. In this case, the questions of dynamo behavior sat in front of our noses, yet remained unanswered for years. The MDE has already increased our knowledge of dynamo flows, and will undoubtedly uncover reasons and explanations for magnetic dynamo behavior in astrophysical bodies, such as the Sun and the Earth. We may know what makes the northern skies shimmer reds and greens and blues, but we have yet to understand why they shimmer. That is what the MDE sets out to answer.
*phew*
Now i've only an exam (french), and three finals left.
Hopefully the weather will warm up.
And hopefully I'll secure a lab job before summer's end.
There's a chance I might sign up for a sublet today, too.
If you feel like reading a boring paper, or seeing what I was up to...
---
In Madison, Wisconsin, if you drive out of the city a few miles on a cold winter’s night and look towards the northern sky, you might be blessed with the spectacular sight of the shimmering red and green flames of the Aurora Borealis. Billions of electrons zoom along the Earth’s magnetic field lines to the poles, violently colliding with gas atoms in the Ionosphere, and releasing light in an awe-inspiring display the entire northern hemisphere can observe . Similarly, if you drive a few miles out of the city and go down underneath the expansive cornfields, you might find the underground bunker used by the Madison Dynamo Experiment to understand the very same magnetic field that gives you the Northern Lights .
It is a well-known fact that the Earth, Sun, and other astrophysical bodies have strong magnetic fields. What is not understood is how these fields developed, nor why they behave the way they do. Professor Cary Forest leads the investigative research project at the University of Wisconsin-Madison in what is called the Madison Dynamo Experiment (MDE). Together with his team of researchers, they hope to answer the many important questions concerning dynamos that remain unsolved: How fast do naturally occurring magnetic fields grow? When do they stop growing? What causes them to stop growing? What types of flow are required for a laminar dynamo (a phenomenon dealing with magnetic fields in astrophysical bodies), and what are the effects of turbulence in a dynamo? These questions are answered in the three major stages of the MDE, that is, the theory, the water vessel experiment, and the sodium vessel experiment.
The primary cause of the strong magnetic fields in astrophysical bodies is generally accepted to be the “dynamo effect.” To begin, a young or forming celestial body (such as a planet or star) has a weak magnetic field. Its interior is made up of various flowing materials or plasmas; in the case of the Earth, this would be molten iron. The flowing of these conducting fluids causes the magnetic field lines to bend and, more importantly, stretch, thereby amplifying the total magnetic field. If the resulting magnetic field continues to grow, eventually it will affect and control the flow of the conducting fluids. Without the need for the original magnetic field, you now have a self-sustained dynamo.
The theory behind the growth and behavior of the magnetic dynamo is described by the appropriately named ‘dynamo equation’, as well as a combination of the magnetic induction equation and the Navier-Stokes equation. The dynamo equation is derived directly from three fundamental laws (Ampère’s Law, Ohm’s Law, and Faraday’s Law), and is essentially a watered down form of the Navier-Stokes equation, with a term included to represent the Lorenz force. The Navier-Stokes equations are partial differential equations that describe the flow and velocity of incompressible fluids. The development of a solution to these equations applicable to astrophysical dynamos was one of the most difficult (and important) obstacles in the theory leading up to UW-Madison dynamo research.
To solve this, the assumption was made that the original magnetic field was weak (and thus the Lorenz force), thereby decoupling the two equations. Consequentially, the linear partial differential equation could be solved.
Two vital discoveries were made from this accomplishment. The first addresses the problem of magnetic field diffusion. Although it may seem obvious, it’s drastically important that the flows in the fluid stretch and bend the magnetic field quicker than they can deteriorate away. As the ratio of energy transfer to diffusion is fundamental to understanding and improving dynamos, it is given a specific name: the magnetic Reynolds number, Rm. The value of this number is directly dependant on the material’s viscosity, conductivity, and size. As the Reynolds number increases for a fluid, the growth rate of the magnetic field grows larger. Since growth is a rate of change, it has a critical number that occurs when it passes from negative to positive values; this number is known as the critical Reynolds number, and determines when the system becomes a dynamo. Any fluid with a Reynolds number below this critical value will have a magnetic field that decays over time, while the magnetic field in a fluid with a Reynolds number greater than the critical value will grow larger.
Obviously, in studying dynamos one of the goals is to reduce the critical Reynolds number in the fluid by as much as possible. This highlights the second discovery made by the MDE: that optimizing the critical Reynolds number can be achieved by adjusting the geometry of the flow to create positive magnetic feedback. The geometry of the flow being studied in the MDE is referred to as the ‘stretch-twist-fold’ dynamo. As the name sounds, the magnetic field lines in the fluid are stretched, twisted up and around, and then folded over like kneaded bread dough. The result is a tangled mess of field lines, plus a new field line situated in the same space and orientation as the original, indicating a new magnetic flux which can then be altered all over again.
Although in the MDE the fluid mechanics of the dynamo seemed to be understood on paper, experimental data to base the final project’s blueprints off of were needed, and as a result the water version of the MDE was built. Under the high velocity (roughly 20 m/s) and high pressure conditions of the spherical dynamo used by Madison researchers, water (due to its viscosity and mass density) behaves almost identically to the conducting fluids planned for the magnetic experiments. Considering its availability and low cost, water was the perfect substitution for studying the motion of particles inside a dynamo and the optimization of apparatus specifications.
In large physical bodies, heat transfers similar to convection pockets provide the kinetic energy needed for a self-sustained dynamo. In the lab setting, propellers are used to imitate this behavior. For the water-based experiments, a stainless steel spherical shell with a diameter of one meter was used . Inside the sphere were two impellors, evenly spaced apart, that rotated in opposite directions. It was fitted with five viewing windows for the purpose of measuring flows and observation. All of the calculations and measurements were done in a spherical coordinate system (as it is the most natural in this setting), and consequently the windows provided a full scan of the apparatus for both r and θ dimensions. A process called Laser Doppler Velocimetry was used to take the actually measurements and collect data on flow geometries. To prevent cavitation at high flow speeds, the windows were also designed to withstand pressures up to 670 kPa.
Unfortunately, a problem arises from using propellers as a substitute for massive astrophysical convection processes. Because higher velocities must be attained, turbulent nonlinearities arise in the flow. In some cases (known as the beta effect), cyclic turbulent patterns in the laminar dynamo can transport magnetic flux out of the system and increase diffusivity, much like the stirring at the edge of a river reduces the net momentum of the flow. The fluid’s conductivity is also reduced, and thus the critical Reynolds number is increased, making it more difficult to produce the dynamo. Whether or not this problem would significantly affect the magnetic-based portion of the research was a major issue for the early stages of the MDE.
Luckily, there was a second, inverse form of turbulence observed that possibly eliminates worry over diffusion due to the beta effect. Tiny helical motions in the flow mirror the system’s behavior and can produce the same stretch-twist-fold on a much smaller scale. These helicies produce current, and thus magnetic field lines. If the flow is carefully tweaked, this form of turbulence (aptly called the alpha effect) can create a positive net magnetic field, and can in fact greatly contribute to forming the desired dynamo.
Water was used primarily to see if the desired flows could be created in the lab using propellers. Fortunately, it was shown that the propellers could be designed specifically to create the desired eigenmode (that is, the least damped, fastest moving eigenmode) and thus the desired dynamo.
The next step was to build the vessel to be used in producing dynamos and studying their behavior. Construction of this apparatus finished just recently with the insertion of the impellors into the vessel. In contrast with the water experiment, the stainless steel sphere has a diameter of 0.5 m, and operates on 150 kW of power (roughly divided between two 75 kW motors for the impellors). It can handle a maximum of 100 kPa of pressure, and peak mean speeds of up to 20 m/s. Instead of windows, eight 5 cm ports were built around the sphere for the insertion of probes.
Several assumptions had to be made in the construction of the second vessel and in making predictions of its results. The first deals with the material that is to be used as the fluid in the dynamo, that is, sodium. If the results from the water experiment are to be extrapolated to the sodium experiment validly, it must be assumed that the flows in the sodium will be identical to those observed in water. This assumption is safe because of the extensive similarities between water and sodium’s physical properties. Both have nearly the same mass density. Also, the viscosity of sodium at 120°C is approximately identical to that of water at 40°C-60°C. Next, it is assumed that the mean velocity field’s shape will not change as power is increased. This is important due to the fact that a larger amount of power will be required for the sodium experiment than was needed for the water experiment. Finally, the other major assumption is that in the UW-Madison lab setting, turbulence is negligible and thus can be ignored. Of course, this is a major assumption, and is justified by the fact that the measured mean quantities are significantly larger then those of the turbulence, and by the lack of a strong Lorenz force inside the lab-based dynamos. Still, it is the one soft spot of the research, and will undoubtedly move to the spotlight as our general understanding of dynamo behavior grows.
With this setup, the flow is expected to achieve Reynolds numbers greater than 100 , which should be significantly higher than the critical Reynolds number for the sodium conducting fluid. Therefore, the system’s magnetic field is expected to grow, until eventually the Lorenz force does become strong and controls the flow geometry of the system. At this point, the conductivity of the fluid is expected to decrease, thus raising the critical Reynolds number. Although this may sound undesirable, it is predicted that when the critical Reynolds number and Reynolds number from the water dynamo experiment intersect, they will flat-line, and the system will become a saturated self-sustained dynamo (a result from the complete solution to the Novier-Stokes/induction equation including the Lorenz force).
In truth, the Madison Dynamo Experiment is just entering into the most exciting phase, where predictions are checked and hopes are met or lost. Now that the sodium dynamo vessel has been built, regular testing and refinement can proceed. It is expected that this phase of the experiment will span close to a decade. Major areas of interest include lowering the critical Reynolds number, understanding saturation more thoroughly, accurately predicting the growth of the dynamo, and refining dynamo theory to account for turbulence. The most important (and interesting) of these is the inclusion of turbulence, as it makes the system significantly more complex and unpredictable. It is very likely that after the beginning sodium tests are complete, the MDE will focus more intensely on furthering our knowledge of turbulent dynamo systems.
In the end, the Madison Dynamo Experiment is a perfect example of research done for the sake of scientific curiosity, and the urge to gain a better understanding of the world around us. In this case, the questions of dynamo behavior sat in front of our noses, yet remained unanswered for years. The MDE has already increased our knowledge of dynamo flows, and will undoubtedly uncover reasons and explanations for magnetic dynamo behavior in astrophysical bodies, such as the Sun and the Earth. We may know what makes the northern skies shimmer reds and greens and blues, but we have yet to understand why they shimmer. That is what the MDE sets out to answer.