Selected Questions -Answers From All Experts Astronomy Forum (Magnetic Helicity)

Question: Can you please explain magnetic helicity - with examples - as applied to the solar environment?   Use any math you need as I have taken advanced calculus! :)


Answer: 

It is good to know you have an advanced calculus background, since to really get at the nitty gritty of helicity - say beyond the "show and tell" stage, one needs some mathematical background. Preferably in differential and integral calculus.


But first things first. It is possible to obtain at least a basic conceptual notion of helicity by doing a simple geometrical exercise which I did in high school and I am sure you probably have as well.

That is, to construct a Moebius strip.  To do this very easily, you can simply cut out a rectangular strip of paper with the rough dimensions shown below:



a[----------------------L---------------------------------------------------]b

b[----------------------L---------------------------------------------------]a


Now take the strip (of length L) and put a kink or half-twist into it about two –thirds from one end, then tape the free ends such that points ab meet ba in that order, i.e.  when taping the ends ensure a to a and b to b.. What you will have is called a Moebius strip.

The Moebius strip has one part twist and one part writhe and this is the fundamental basis of helicity. You can get a pictorial idea by going to:

http://en.wikipedia.org/wiki/M%C3%B6bius_strip


“Magnetic helicity” was probably first introduced by K. Moffat in the late 1950s as a topological invariant that describes the complexity of a magnetic field. Like the pure tolpological helicity, this magnetic helicity also has “twist” and “writhe” components. It is written as a function of the vector potential (A) and the magnetic field (B), and measures the topological linkage of magnetic fluxes (F)

The magnetic helicity H of a magnetic  field B within a volume V is defined:

H =  ò_V  A · B dV

 And the vector potential A satisfies:

B = curl A     e.g. Ñ X A  =  B

Here A is the vector potential. From Taylor’s hypothesis, the above integral is approximately invariant- so the minimum energy configuration is a “constant-alpha” force free field, e.g. curl B = aB

Where  a  is a linear scale factor, such that: a = m_o J/ B

In actual working solar conditions, one prefers a gauge-invariant form of H and this is provided by the “relative helicity” – wherein one subtracts the helicity of some reference field (B_o, e.g. associated with  a= 0) and having the same distribution of the normal component of B on the surface (S).  As we see, with a= 0  then J = 0, B = 0.    This is a current-free field given  curl B = 0.  (e.g. Ñ X B  = a B  = 0)   Another way to put this, is that such a solar configuration represents a "potential field" i.e. which has zero free magnetic energy stored.  Then:

H _r  = ò V     A  B dV  -   ò V_o  A _o B_o dV_o

The  magnetic storage capacity takes on actual values as the field departs from the potential and we get non-zero a  showing a measure of shear.  It is hypothesized that shearing and twisting of the field “injects” helicity and that this may be useful in quantifying: a) how much magnetic free energy becomes available, and b) whether instability can be predicted based on observed indicators of helicity at the level of the photosphere-chromosphere.

For example it may be possible to resolve H _r  observationally into two components such that:

d H _r / dt =  d H _r  [T] / dt    +  d H _r [W] / dt   

where term (1)  on the right side refers to the “twist” and term (2)  to the “writhe” (a topological function of the specific geometry observed)  An illustration depicting acquisition of both forms of helicity is shown in the graphic below:

Images of rising solar loops: On the left side the loop features only internal twist, on the right the loop has writhe as well as twist. (From DeMoulin and Berger,  Solar Physics, 2003)

Theoretically, even if   (d H _r / dt)   = 0, the writhe term is non-zero, hence a residual helicity remains. This is why conservation of helicity defines a specific constraint such that no magnetic field with finite total helicity can relax to a potential field (Demoulin et al, 2006)

We see evidence of the Sun’s magnetic helicity in the solar corona as well as the solar wind that streams past Earth. In eruptive prominences, for example see the one in the attached image, we actually have the graphical detection of the twist and writhe (or helical structure) associated with topological helicity – and which is magnetic helicity in the Sun’s magnetic environment.

Also, if you carefully inspect and study the prominence in the upper right of the image in the link below, you can discern both twist and writhe in the plasma filaments. Evidently then, prominences are capable of transporting magnetic helicity in the solar corona.

http://antwrp.gsfc.nasa.gov/apod/ap060416.html

The magnetic helicity is also visible in the gas filaments of the prominence depicted in the lower left of the image from the link below:

http://antwrp.gsfc.nasa.gov/apod/ap051109.html

Solar eruption, especially coronal mass ejections (CMEs) carry magnetic flux as well as helicity from the Sun. When the erupted magnetic field reaches the Earth it interacts with the magnetosphere, causing magnetic substorms and auroras.

Some recent research also reveals remarkable aspects of magnetic helicity in the solar environment. For example, it seems that magnetic helicity of different signs or polarities (+ or -) can occur, depending on which hemisphere of the Sun it’s measured.

Specifically, the sign of helicity will be positive or negative, depending on what is known as the “hemispheric helicity rule.” That is, the force-free  a  characterizing each active region will have a tendency to be (+) in the southern solar hemisphere, (-) in the northern solar hemisphere.

When you think about it, this makes eminent sense. (Think of the Coriolis force causing a preferred sign or handedness, relative to convective flows in the northern and southern hemispheres of a planet like Earth) If there is a preferred “handedness” (or chirality) associated with magnetic flux, it would be expected to exhibit a different sign in each hemisphere.

Observations confirm that this sign asymmetry exists throughout the solar atmosphere: in the corona, the solar wind and the photosphere.

Water Sources Rapidly Vanishing Owing To Food Waste

Seven years ago a  study released  by the UN Food and Agriculture Organization (FAO). showed roughly 1.3 billion tons of food was either lost or wasted globally due to inefficiencies throughout the food supply chain.  According to the report, industrialized and developing countries wasted or lost roughly the same amount of food each year – 670 million and 630 million tons respectively. But while rich countries wasted food primarily at the level of the consumer, the main issue for developing countries was food lost due to weak infrastructure – including poor storage, and inadequate processing and packaging facilities that lack the capacity to keep produce fresh.

 At the time it had been made evident that food losses mean lost income for small farmers and higher prices for poor consumers in developing countries. Now we also know the exent to which food waste and losses mean wasted water resources as well. 

According to a new report  published in Plos One Americans alone wasted over 25 percent of their food between 2007 and 2014.   Researchers at the US Department of Agriculture analyzed eight years of food dat  to see where food is wasted and also what members of the public say they do at meal times.  About 150,000 tons of food is tossed out in US households each day, equivalent to about a third of the daily calories that each American consumes. Fruit and vegetables were the most likely to be thrown out, followed by dairy and then meat.

This waste was also found to have an environmental toll, with the volume of discarded food equivalent to the yearly use of 30 m acres of land and 4.2 trillion gallons of irrigated water. Rotting food not only denies nutrition to a vast population that needs it but also clogs up landfills and releases methane, a powerful greenhouse gas.

Ironically, the study found that the healthiest Americans were the most wasteful, because of their high consumption of fruits and vegetables, which are frequently thrown out. Fruit and vegetables require less land to grow than  other foods, such as meat, but require a large amount of water and pesticides. 

In terms of  breaking down the quantitative specifics.  the uneaten fruits and vegetables accounted for nearly 2.3 trillion gallons of water wasted, i.e. that was used to grow them (1.3 trillion for fruits, 1.0 trillion for veggies). This is serious given that a ‘State of the World’ report (2000, pp. 46-47), warned that ever increasing water deficits will likely spark “water wars” by 2025 - with more than 500 million people in a condition of water deprivation.

 Fruits and vegetables  also accounted for 39 percent of the total food wasted by weight.

In addition, farmers used 1.8 billion pounds of nitrogen fertilizer - which adversely affects marine and terrestrial ecosystems. Also, 1.5 billion pounds of phosphorus fertilizer which can feed algal blooms that are inimical to fish, e.g. 


Also excessive use was made of potash fertilizer (2.3 billion pounds) used annually to grow the wasted crops.  Those wasted crops also required  780 million pounds of carcinogenic pesticides - i.e. to protect food that never passed anyone's lips but is now wreaking havoc with breast, prostate, thyroid, liver and colon cancers.

Lisa Jahns, a nutritionist at USDA and co-author of the study, said:

“We need a simultaneous effort to increase food quality as well as reduce food waste. We need to put both of those things out.”

Jahns’s study recommends educating consumers on fruit and vegetable storage in order to reduce food waste.  This is an excellent idea, given if we are going to expose our bodies to carcinogenic pesticides in growing the veggies, we ought to be sure they are consumed not tossed out.

Jahn added:

“Consumers aren’t connecting the dots, [and] they don’t see the cost when they throw food in the trash. At the same time, we don’t want to undermine legitimate food safety concerns and we need to be aware it’s not just the cost of food that’s the issue. It’s the time and energy required to prepare and store food, which often isn’t a priority in a busy household.”

The report, by the Center for Biological Diversity, found that just four of the 10 largest grocery chains in the U.S. have specific food waste reduction commitments. A further four out of the 10 don’t prevent the waste of food considered too cosmetically “imperfect” to sell. Walmart achieved the highest grade, a B, while Aldi US was the worst with an 'F. Trader Joe’s, Target and Whole Foods all did poorly, with a D.

Jordan Figueiredo, who runs the “Ugly” Fruit and Veg Campaign and collaborated on the report told The Guardian UK

"There is just so much wasted food at so many levels,”Supermarkets usually get a free pass for doing anything to reduce wasted food or donate extra food. There is so much more they can and should do to reduce wasted food, especially given that they are essentially the gatekeepers for our food.”

Perhaps the most severe indictment for food waste comes against the healthy eaters.  Indeed, the study found "the higher the diet quality the more food was wasted"

Specifically, Americans in the bottom quintile of income wasted an average of 295 grams of food per day, while those in the top  quintile wasted an average of 535 grams of food each day, or nearly 81 percent more. 

Maybe this is one more reason we need to ditch the healthy eating craze given we already know it leads to higher medical costs too, e.g.

http://brane-space.blogspot.com/2017/01/healhy-eating-living-leads-to-massive.html

See also:

https://www.sciencedaily.com/releases/2018/04/180418141508.htm

 

Footnote:

Want a more comprehensive tally of waste? In “Junk Planet”, Robert Burrowes  provides a comprehensive description of the degradation of the atmosphere, oceans, waterways, groundwater, and soil as well as the modern pollution of antibiotic waste, genetic engineering, nanowaste, space junk, military waste and nuclear, a description of a planet degraded by pollution impacting our bodies and health as well as the planet’s future.