Mean Field Theory and Solar Dynamo Modeling

In a recent post, I talked about the characteristics of the sun’s 11 and 22 year cycles, the observed laws which describe the behavior of the sunspot cycle, how proxy data is used to reconstruct a record of solar cycles of the past, Grand Solar Maxima and Minima, the relationship between Total Solar Irradiance (TSI) and the sunspot cycle, and the relevance of these factors to earth’s climate system. In a follow up post, I went over the structure of the sun, and some of the characteristics of each layer, which laid the groundwork for my last post, in which I explained the solar dynamo: the physical mechanism underlying solar cycles.

Before elaborating on the sun’s role in climate change in the installment following this one, I’ll be going over an approach called “Mean Field Theory” in this installment, which dynamo theorists and other scientists sometimes use to make the modelling of certain systems more manageable. As was the case with part III, this may be a bit more technical than most of my subscribers are accustomed to, but I think the small subset of readers with the tools to digest it will appreciate it. And to be perfectly blunt, writing this was not just about my subscribers. I wanted to do it. It was an excuse for me to dig more deeply into something that has been going on in modern stellar astrophysics that I thought was interesting. The fact that it happened to be tangentially related to my series on climate science was a mere convenience. Anyone wanting to avoid the math and/or to cut to the chase with respect to the effects of solar cycles on climate change might want to skip ahead to part V, or perhaps just read only the text portions of this post. However, for those who don’t mind a little bit of math, I present to you the following:

Mean Field Theory

One approach by which scientists and mathematicians can simplify the models describing large complex systems stochastic effects is called Mean Field Theory (Schrinner 2005). This involves subsuming multiple complicated interactions between different parts of a system into a single averaged effect. In this way, multi-body problems, (which are notoriously difficult to solve even with numerical approximation methods with supercomputers), can be reduced to simpler single body problems. For instance, the velocity field u and magnetic field B could each be broken up into two separate terms: A mean term (u0 and B0 respectively), and a fluctuating term (u’ and B’ respectively), whereby the mean terms are taken as averages over time and/or space depending on what is appropriate to the system being modeled.

In other words, u = u0 + u’ and B = B0 + B’, where (by definition) the average velocity <u> = u0, the average magnetic field <B> = B0, because u0 and B0 are the mean field terms, and the average of the fluctuating terms <B’> and <u’> are both equal to zero by definition. The angled brackets simply denote a suitable average of the term they enclose (again taken over time and/or space as deemed appropriate by the scientist or mathematician). The reason the fluctuating terms average out to zero is because the mean field terms are defined as the average of the entire field, so by definition, the only way that can be true is if the fluctuating terms average out to zero.

Using the vector calculus identity × ( × B) = ∇(∇·B) + 2B, and the fact that ·B = 0 by Gauss’s Law for magnetic fields, the induction equation, ∂B/∂t  = × (u × B − η × B) from the previous section can also be expressed as ∂B/∂t = η2B + × (u × B), where 2B is called the Laplacian operator of the magnetic field B.

Plugging in our mean field equations u and B into this form of the induction equation:

∂B0/∂t + ∂B’/∂t = η2B0 + η2B’+

× (u0× B0) + × (u0 × B’) +

× (u’ × B0) + × (u’ × B’)

Now we take the average of both sides:

∂<B0>/∂t + ∂<B’>/∂t = η2<B0> + η2 <B’> + 

× <u0× B0> + × <u0 × B’> +

× <u’ × B0> + × <u’ × B’>

However, we’ve already know that <B0> = B0, <u0> = u0, and <u’> = < B’> = 0, so this can be simplified:

∂B0/∂t = η2B0 + × <u0× B0> + × <u’ × B’>.

The × <u’ × B’> is typically then replaced with a term × ε, where ε is called the mean electromotive force (Radler 2007). This yields the Mean Field Induction Equation:

∂B0/∂t = η2B0 + × <u0× B0 + ε>

Although many details of the theory are still being worked out, models based on the solar dynamo mechanism are consistent with the periodicity of the solar cycle, Hale’s law (the opposing magnetic polarity of sunspots above and below the solar equator and the alternation of polarity in successive 11 year cycles), as well as both Sporer’s and Joy’s laws (the apparent migration of sunspots towards the equator as a cycle progresses, as well as their tilt), which together produce the observed sunspot butterfly diagrams I talked about here.

Some models can even simulate variations in amplitude from one cycle to the next, but the precise manner in which Grand Solar Maxima and Minima emerge is still being worked out. Consequently, our models’ ability to reliably and accurately forecast them is currently still limited. Methods have been developed for estimating sunspot number and solar activity of a cycle’s solar maxima from observations of the poloidal field strength of the preceding cycle’s solar minima (Schatten 1978, Svalgaard 2005). In addition to only providing information on the solar maxima immediately after the minima being measured, this approach is also limited by the fact that our poloidal field measurements only go back a few cycles, and because the poloidal magnetic fields during solar minima are difficult to measure reliably, because they are weak and have radial as well as meridional components.

Other researchers have focused on kinematic flux transport solar dynamo models which, in addition to differential rotation, include the effects of meridional flow in the convective envelope, whereby the poloidal magnetic field is regenerated by the decay of the bipolar magnetically active regions subsequent to their emergence at the solar surface (Dikpati 1999, Dikpati 2006, Choudhuri 2007). Active regions are the high magnetic flux regions at which sunspots emerge.

Image  by Andres Munoz-Jaramillo (check out his fantastic presentations on solardynamo.org).

This meridional flow sets the period of the cycle, the strength of the poloidal field, and the amplitude of the solar maximum of the subsequent cycle. However, estimates of meridional flow velocities prior to 1996 are highly uncertain (Hathaway 1996). All of these models have been criticized by peers of their proponents. A concise summary of the blow by blow can be viewed here.

As for Grand Solar Maxima and Minima, no comprehensive theory has yet emerged on how they arise and decay, let alone a scientific consensus. However, certain constraints have been identified. There is evidence that the dynamo cycle does continue in some modified form during Maunder-type minima periods. The idea is that the dynamo enters Grand Maxima and Minima by way of chaotic and/or stochastic processes. In the case of Grand Maxima, the dynamo also exits that state via stochastic processes. In the case of Grand Minima, on the other hand, the dynamo then gets “trapped” in this state, but eventually gets out of it via deterministic internal processes (Usoskin 2007). It is also thought that the polarity of the sun’s toroidal magnetic field may lose its equatorial anti-symmetry during such minima, and instead become symmetric (Beer, Tobias and Weiss 1998).

Truly fantastic long term predictive power for solar cycles probably won’t be achieved until poloidal magnetic field generation is better understood, which will likely include improvements in flux transport models, and a more complete characterization of the statistical properties of bipolar magnetic regions (BMRs). For a comprehensive overview of the current state of Solar Dynamo Models and their predictive strengths and limitations, see Charbonneau 2010.

In the next installment, I’ll explain how all of this relates to climate change on earth, and address the elephant in the room: “are solar variations responsible for the current global warming trend?”

Related Articles:

References:

Beer, J., Tobias, S., & Weiss, N. (1998). An active Sun throughout the Maunder minimum. Solar Physics181(1), 237-249.

Charbonneau, P. (2010). Dynamo models of the solar cycle. Living Reviews in Solar Physics7(1), 1-91.

Choudhuri, A. R., Chatterjee, P., & Jiang, J. (2007). Predicting solar cycle 24 with a solar dynamo model. Physical review letters98(13), 131103-131103.

Coriolis, G. G. (1835). Théorie mathématique des effets du jeu de billard. Carilian-Goeury.

Dikpati, M., & Charbonneau, P. (1999). A Babcock-Leighton flux transport dynamo with solar-like differential rotation. The Astrophysical Journal518(1), 508.

Dikpati, M., De Toma, G., & Gilman, P. A. (2006). Predicting the strength of solar cycle 24 using a flux‐transport dynamo‐based tool. Geophysical research letters33(5).

Hathaway, D. H. (1996). Doppler measurements of the sun’s meridional flow. The Astrophysical Journal460, 1027.

Rädler, K. H., & Rheinhardt, M. (2007). Mean-field electrodynamics: critical analysis of various analytical approaches to the mean electromotive force. Geophysical & Astro Fluid Dynamics101(2), 117-154.

Schatten, K. H., Scherrer, P. H., Svalgaard, L., & Wilcox, J. M. (1978). Using dynamo theory to predict the sunspot number during solar cycle 21. Geophysical Research Letters5(5), 411-414.

Schrinner, M., Rädler, K. H., Schmitt, D., Rheinhardt, M., & Christensen, U. (2005). Mean‐field view on rotating magnetoconvection and a geodynamo model. Astronomische Nachrichten326(3‐4), 245-249.

Svalgaard, L., Cliver, E. W., & Kamide, Y. (2005). Sunspot cycle 24: Smallest cycle in 100 years?. GEOPHYSICAL RESEARCH LETTERS32, L01104.

Usoskin, I. G., Solanki, S. K., & Kovaltsov, G. A. (2007). Grand minima and maxima of solar activity: new observational constraints. Astronomy & Astrophysics471(1), 301-309.

A Compilation of Studies and Articles on GE Food Safety and the Scientific Consensus

The following is a list of studies and articles on GE food safety and the scientific consensus, which I’ve complied for convenient access. It will be updated periodically.

waitingtofindharm

A practical Introduction to Vectors

The mathematical concept of a “vector” is ubiquitous in the realms of physics, engineering and applied mathematics. Typically, the concept is first introduced (usually in a first semester physics course or perhaps a trigonometry or calculus course) as a quantity with both a magnitude and a direction, and is usually represented as a line segment with an arrow at the end of it (like a ray).

Often, these introductory treatments will distinguish what a vector is by contrasting it to what a scalar is. A scalar is just a quantity (albeit possibly with some units of measurement), such as length, weight, mass, speed, an amount of money, or a frequency. On the other hand, examples of vectors would be things like velocity, acceleration, force, momentum, and position (relative to some set of coordinates). In other words, they have a direction as well as a magnitude.

NASA

Although vectors can in principle be adapted to any coordinate system, for the sake of simplicity, we will presuppose a Cartesian coordinate system in either 1, 2, or 3 dimensions. In this representation, we’ll have a horizontal x axis and a vertical y axis for 2D, and for 3D we’ll have the y axis horizontal, the z axis vertical, and the x axis coming out of the page at the reader. This is just a matter of historical convention. Occasionally one might see the z axis being used as the one coming out of the page, but most of my pictorial examples involved the former and not the latter. This is a type of coordinate system with what’s called an “orthonormal” basis. I’ll cover the concept of a basis in a later post, but in practical terms, orthonormal just means that the coordinate axes are perpendicular to one another. Any points or vectors in the space are constructed by adding linear combinations of  unit vectors (magnitude equal to 1) in the x, y and z directions.

A vector can be represented by a letter with a little arrow on top of it. but since it’s a PITA to write them that way in this format, I’ll just use capital letters to name vectors for now. Suppose we have a vector, A. There are a number of common notational conventions for representing it.

A = [a1, a2, a3],

or

\vec{A}=(A_x)\hat{i}+(A_y)\hat{j}+(A_z)\hat{k},

where the i, j and k with the little hats each represent a unit vector (a vector of unit length) in one of the component directions (x, y, and z respectively in this case), and the a1, a2 and a3, (or A_x, A_y and A_z) symbols represent the “components” of the vector in the x, y and z directions respectively. I will use the A = [a1, a2, a3] format for convenience.

So, for example, supposing you were traveling at 10 m/s in the x-direction. Then your velocity vector v = [10, 0, 0] m/s, since you’re not moving in the y or z directions at all.

Vectors can be added and subtracted just like scalar quantities, but there is a procedure to it. The process is to add or subtract each component of the vector.

For example, if you had a vector, A = [a1, a2, a3], and another vector, B = [b1, b2, b3].

Then A + B = [a1 + b1, a2 + b2, a3 + b3].

Similarly, A – B = [a1 – b1, a2 – b2, a3 – b3].

Let’s try a more concrete example. Supposing your positive x component corresponded to East, your positive y component corresponded to North, and your positive z compodent corresponded to up in the air. Supposing you had two cars driving in an open field. Car A has a velocity v_a = [40, 30, 0] km/hr, (that’s diagonal motion of 40 km/hr east and 30 km north) relative to an observer on the ground, and car B is moving at a velocity v_b = [0, -40, 0] km/hr.

If you wanted to know A’s velocity from the perspective of B, then you’d subtract the velocity of B from the velocity of A.

v_a – v_b = [ 40 – 0, 30 – (-40), 0 – 0] km/hr = [40, 70, 0].

The following Khan Academy video explains some more examples of this:

https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/vectors/v/vector-introduction-linear-algebra

Something that is typically not mentioned in such introduction (mostly for the sake of expediency), is the fact that this definition of a vector is really only a special case of a much broader-reaching concept rife with many other special cases, each comprising their own unique attributes and applications. Later on, it is customary for the student to learn a more generalized concept of vector “spaces,” whereby a vector space is defined by a list of rules by which the elements of a space must abide in order to qualify as a vector space. This is traditionally taught in a first course on linear algebra, along with concepts such as “rank,” “dimension,” “linear independence,” and a “basis,” and opens the door to a lot of vector spaces which wouldn’t necessarily fit the description of a vector that students are typically taught in introductory physics. These included things like vector “fields” and topological vector spaces, of which metric spaces are a subset, of which normed vector spaces are a subset, of which inner product spaces and banach spaces are subsets, of which Hilbert spaces are a subset (etc). The latter, (Hilbert Spaces), are ubiquitous in quantum mechanics and quantum field theory for example. There may also come a point at which a student comes to learn that some vectors are also a subset of a class of mathematical objects known as “tensors.” Not all vectors are tensors, (but all first order tensors are actually vectors), and not all scalars are tensors, (but all zeroth order tensors are scalars).

330px-Mathematical_Spaces.svg

 

In actuality, the hierarchy is a bit more involved than depicted here, and understanding it would require covering some concepts such as “completeness” from a branch of math known as functional analysis, which I will not attempt to cover here.

 

800px-Mathematical_implication_diagram_eng

 

Don’t worry about not knowing what distinguishes these types of spaces from one another right now. The point is to state up front that there multiple levels to this concept of vectors, instead of simply conveniently neglecting to mention that until it can’t be put off any longer, as is often done in the earlier college courses. So rather than getting lost down that rabbit hole, just know there are a few basic operations and notational conventions to understand for vectors in the sense of a “quantity possessing a magnitude and a direction,” but that those are but a special case of a broader and very useful mathematical concept that has applications in various scientific sub-disciplines.

In a subsequent lesson, I can go over vector multiplication (i.e. dot products/inner products and cross products), scalar projections, vector projections, vectors as functions of independent variables, vector calculus, the concept of a vector space, and various useful operations involving vectors and vector functions.