# Modeling Earth’s Energy Balance (Zero-D) (Equilibrium)

#### February 22, 2017

For conservation of energy, the short-wave solar energy absorbed by the Earth equals the long-wave outgoing radiation.

Energy and matter can’t just disappear. Energy can change from one form to another. As a thrown ball moves upwards, its kinetic energy of motion is converted to potential energy due to gravity. So we can better understand systems by studying how energy (and matter) are conserved.

## Energy Balance for the Earth

Let’s start by considering the Earth as a simple system, a sphere that takes energy in from the Sun and radiates energy off into space.

### Incoming Energy

At the Earth’s distance from the Sun, the incoming radiation, called insolation, is 1367 W/m2. The total energy (wattage) that hits the Earth (Ein) is the insolation (I) times the area the solar radiation hits, which is the area a cross section of the Earth (Acx).

$E_{in} = I \times A_{cx}$

Given the Earth’s radius (rE) and the area of a circle, this becomes:

$E_{in} = I \times \pi (r_E)^2$

### Outgoing Energy

The energy radiated from the Earth is can be calculated if we assume that the Earth is a perfect black body–a perfect absorber and radiatior of Energy (we’ve already been making this assumption with the incoming energy calculation). In this case the energy radiated from the planet (Eout) is proportional to the fourth power of the temperature (T) and the surface area that is radiated, which in this case is the total surface area of the Earth (Asurface):

$E_{out} = \sigma T^4 A_{surface}$

The proportionality constant (σ) is: σ = 5.67 x 10-8 W m-2 K-4

Note that since σ has units of Kelvin then your temperature needs to be in Kelvin as well.

Putting in the area of a sphere we get:

$E_{out} = \sigma T^4 4 \pi r_{Earth}^2$

### Balancing Energy

Now, if the energy in balances with the energy out we are at equilibrium. So we put the equations together:

$E_{in} = E_{out}$

$I \times \pi r_{Earth}^2 = \sigma T^4 4 \pi r_{Earth}^2$

cancelling terms on both sides of the equation gives:

$I = 4 \sigma T^4$

and solving for the temperature produces:

$T = \sqrt{\frac{I}{4 \sigma}}$

Plugging in the numbers gives an equilibrium temperature for the Earth as:

T = 278.6 K

Since the freezing point of water is 273K, this temperature is a bit cold (and we haven’t even considered the fact that the Earth reflects about 30% of the incoming solar radiation back into space). But that’s the topic of another post.

Citing this post: Urbano, L., 2017. Modeling Earth's Energy Balance (Zero-D) (Equilibrium), Retrieved February 23rd, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Elements by Their Uses

#### January 19, 2017

Keith Enevoldsen has an excellent version of the periodic table that has some of the more popular uses of the elements on them.

Hat tip to Ms. Douglass for this link.

Citing this post: Urbano, L., 2017. Elements by Their Uses, Retrieved February 23rd, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# The Essential Biopolymers

#### January 10, 2017

My notes on the four macromolecules that are essential to life as we know it: proteins, fats (lipids), carbohydrates (saccharides), and nucleic acids.

Carbohydrates/saccharides.

Fats/lipids.

Nucleic Acids (DNA and RNA).

Proteins.

Citing this post: Urbano, L., 2017. The Essential Biopolymers, Retrieved February 23rd, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Seeing the Rock Cycle in the Ozarks

#### September 25, 2016

On this year’s trip to the Current River with the Middle School we were able to see outcrops of the three major types of rocks: igneous, metamorphic, and sedimentary.

## Igneous Rocks

Beautiful, pink granite at Elephant Rocks State Park.

We stopped by Elephant Rocks State Park on the way down to the river to check out the gorgeous pink granite that makes up the large boulders. The coarse grains of quartz (translucent) interbedded with the pink orthoclase crystals make for an excellent example of a slow-cooling igneous rock.

## Metamorphic Rocks

The Prairie Creek waterfall pool.

On the second day out on the canoes we clambered up the rocks in the Prairie Creek valley to see jump into the small waterfall pool. The rocks turned out to look a lot like the granite of Elephant Rocks if the large crystals had been heated up and deformed plastically. This initial stage of the transformation allowed me to talk about metamorphic rocks althought we’ll see some much more typical samples when we get back to the classroom.

Prairie Creek rocks.

## Sedimentary Rocks

Limestone bluffs along the Current River.

We visited a limestone cave on the third day, although we’ve been canoeing through a lot of limestone for on the previous two days. This allowed us to talk about sedimentary rocks: their formation in the ocean and then uplift via tectonic collisions.

## The Rock Cycle

Diagram of a convergent tectonic margin used to illustrate the rock cycle.

Back at camp, we summarized what we saw with a discussion of the rock cycle, using a convergent plate margin as an example. Note: sleeping mats turned out to be excellent models for converging tectonic plates.

Note to self: It might make sense to add extra time at the beginning and end of the trip to do some more geology stops. Johnson Shut-Inns State Park is between Elephant Rocks and Eminence, and we saw a lot of interesting sedimentary outcrops on the way back to school as we headed up to Rolla.

Citing this post: Urbano, L., 2016. Seeing the Rock Cycle in the Ozarks, Retrieved February 23rd, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Experimenting with Genetic Algorithms

#### September 13, 2016

Genetic algorithm trying to find a series of four mathematical operations (e.g. -3*4/7+9) that would result in the number 42.

I’m teaching a numerical methods class that’s partly an introduction to programming, and partly a survey of numerical solutions to different types of problems students might encounter in the wild. I thought I’d look into doing a session on genetic algorithms, which are an important precursor to things like networks that have been found to be useful in a wide variety of fields including image and character recognition, stock market prediction and medical diagnostics.

The ai-junkie, bare-essentials page on genetic algorithms seemed a reasonable place to start. The site is definitely readable and I was able to put together a code to try to solve its example problem: to figure out what series of four mathematical operations using only single digits (e.g. +5*3/2-7) would give target number (42 in this example).

The procedure is as follows:

• Initialize: Generate several random sets of four operations,
• Test for fitness: Check which ones come closest to the target number,
• Select: Select the two best options (which is not quite what the ai-junkie says to do, but it worked better for me),
• Mate: Combine the two best options semi-randomly (i.e. exchange some percentage of the operations) to produce a new set of operations
• Mutate: swap out some small percentage of the operations randomly,
• Repeat: Go back to the second step (and repeat until you hit the target).

And this is the code I came up with:

genetic_algorithm2.py

''' Write a program to combine the sequence of numbers 0123456789 and
the operators */+- to get the target value (42 (as an integer))
'''

'''
Procedure:
1. Randomly generate a few sequences (ns=10) where each sequence is 8
charaters long (ng=8).
2. Check how close the sequence's value is to the target value.
The closer the sequence the higher the weight it will get so use:
w = 1/(value - target)
3. Chose two of the sequences in a way that gives preference to higher
weights.
4. Randomly combine the successful sequences to create new sequences (ns=10)
5. Repeat until target is achieved.

'''
from visual import *
from visual.graph import *
from random import *
import operator

# MODEL PARAMETERS
ns = 100
target_val = 42 #the value the program is trying to achieve
sequence_length = 4  # the number of operators in the sequence
crossover_rate = 0.3
mutation_rate = 0.1
max_itterations = 400

class operation:
def __init__(self, operator = None, number = None, nmin = 0, nmax = 9, type="int"):
if operator == None:
n = randrange(1,5)
if n == 1:
self.operator = "+"
elif n == 2:
self.operator = "-"
elif n == 3:
self.operator = "/"
else:
self.operator = "*"
else:
self.operator = operator

if number == None:
#generate random number from 0-9
self.number = 0
if self.operator == "/":
while self.number == 0:
self.number = randrange(nmin, nmax)
else:
self.number = randrange(nmin, nmax)
else:
self.number = number
self.number = float(self.number)

def calc(self, val=0):
# perform operation given the input value
if self.operator == "+":
val += self.number
elif self.operator == "-":
val -= self.number
elif self.operator == "*":
val *= self.number
elif self.operator == "/":
val /= self.number
return val

class gene:

def __init__(self, n_operations = 5, seq = None):
#seq is a sequence of operations (see class above)
#initalize
self.n_operations = n_operations

#generate sequence
if seq == None:
#print "Generating sequence"
self.seq = []
self.seq.append(operation(operator="+"))  # the default operation is + some number
for i in range(n_operations-1):
#generate random number
self.seq.append(operation())

else:
self.seq = seq

self.calc_seq()

#print "Sequence: ", self.seq
def stringify(self):
seq = ""
for i in self.seq:
seq = seq + i.operator + str(i.number)
return seq

def calc_seq(self):
self.val = 0
for i in self.seq:
#print i.calc(self.val)
self.val = i.calc(self.val)
return self.val

def crossover(self, ingene, rate):
# combine this gene with the ingene at the given rate (between 0 and 1)
#  of mixing to create two new genes

#print "In 1: ", self.stringify()
#print "In 2: ", ingene.stringify()
new_seq_a = []
new_seq_b = []
for i in range(len(self.seq)):
if (random() < rate): # swap
new_seq_a.append(ingene.seq[i])
new_seq_b.append(self.seq[i])
else:
new_seq_b.append(ingene.seq[i])
new_seq_a.append(self.seq[i])

new_gene_a = gene(seq = new_seq_a)
new_gene_b = gene(seq = new_seq_b)

#print "Out 1:", new_gene_a.stringify()
#print "Out 2:", new_gene_b.stringify()

return (new_gene_a, new_gene_b)

def mutate(self, mutation_rate):
for i in range(1, len(self.seq)):
if random() < mutation_rate:
self.seq[i] = operation()

def weight(target, val):
if val <> None:
#print abs(target - val)
if abs(target - val) <> 0:
w = (1. / abs(target - val))
else:
w = "Bingo"
print "Bingo: target, val = ", target, val
else:
w = 0.
return w

def pick_value(weights):
#given a series of weights randomly pick one of the sequence accounting for
# the values of the weights

# sum all the weights (for normalization)
total = 0
for i in weights:
total += i

# make an array of the normalized cumulative totals of the weights.
cum_wts = []
ctot = 0.0
cum_wts.append(ctot)
for i in range(len(weights)):
ctot += weights[i]/total
cum_wts.append(ctot)
#print cum_wts

# get random number and find where it occurs in array
n = random()
index = randrange(0, len(weights)-1)
for i in range(len(cum_wts)-1):
#print i, cum_wts[i], n, cum_wts[i+1]
if n >= cum_wts[i] and n < cum_wts[i+1]:

index = i
#print "Picked", i
break
return index

def pick_best(weights):
# pick the top two values from the sequences
i1 = -1
i2 = -1
max1 = 0.
max2 = 0.
for i in range(len(weights)):
if weights[i] > max1:
max2 = max1
max1 = weights[i]
i2 = i1
i1 = i
elif weights[i] > max2:
max2 = weights[i]
i2 = i

return (i1, i2)

# Main loop
l_loop = True
loop_num = 0
best_gene = None

##test = gene()
##test.print_seq()
##print test.calc_seq()

# initialize
genes = []
for i in range(ns):
genes.append(gene(n_operations=sequence_length))
#print genes[-1].stringify(), genes[-1].val

f1 = gcurve(color=color.cyan)

while (l_loop and loop_num < max_itterations):
loop_num += 1
if (loop_num%10 == 0):
print "Loop: ", loop_num

# Calculate weights
weights = []
for i in range(ns):
weights.append(weight(target_val, genes[i].val))
# check for hit on target
if weights[-1] == "Bingo":
print "Bingo", genes[i].stringify(), genes[i].val
l_loop = False
best_gene = genes[i]
break
#print weights

if l_loop:

# indicate which was the best fit option (highest weight)
max_w = 0.0
max_i = -1
for i in range(len(weights)):
#print max_w, weights[i]
if weights[i] > max_w:
max_w = weights[i]
max_i = i
best_gene = genes[max_i]
##        print "Best operation:", max_i, genes[max_i].stringify(), \
##              genes[max_i].val, max_w
f1.plot(pos=(loop_num, best_gene.val))

# Pick parent gene sequences for next generation
# pick first of the genes using weigths for preference
##        index = pick_value(weights)
##        print "Picked operation:  ", index, genes[index].stringify(), \
##              genes[index].val, weights[index]
##
##        # pick second gene
##        index2 = index
##        while index2 == index:
##            index2 = pick_value(weights)
##        print "Picked operation 2:", index2, genes[index2].stringify(), \
##              genes[index2].val, weights[index2]
##

(index, index2) = pick_best(weights)

# Crossover: combine genes to get the new population
new_genes = []
for i in range(ns/2):
(a,b) = genes[index].crossover(genes[index2], crossover_rate)
new_genes.append(a)
new_genes.append(b)

# Mutate
for i in new_genes:
i.mutate(mutation_rate)

# update genes array
genes = []
for i in new_genes:
genes.append(i)

print
print "Best Gene:", best_gene.stringify(), best_gene.val
print "Number of iterations:", loop_num
##



When run, the code usually gets a valid answer, but does not always converge: The figure at the top of this post shows it finding a solution after 142 iterations (the solution it found was: +8.0 +8.0 *3.0 -6.0). The code is rough, but is all I have time for at the moment. However, it should be a reasonable starting point if I should decide to discuss these in class.

Citing this post: Urbano, L., 2016. Experimenting with Genetic Algorithms, Retrieved February 23rd, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Carbon in the Ground and Free Oxygen in the Air

#### September 6, 2016

A couple new article relevant to our study of Earth History.

Carbon

Image by Rajdeep Dasgupta, via pyhs.org.

Research on the high pressure and temperature conditions at the Earth’s core suggest that most of the carbon in the early Earth should have either boiled off into space or been trapped by the iron in the core. So where did all the carbon necessary for life come from? They suggest from the collision of an embryonic planet (with lots of carbon in its upper layers) early in the formation of the solar system.

Free Oxygen

Typical surface view of purple mat surface at Middle Island showing purple filaments. Some white filaments can also be observed. Image from Thunder Bay Sinkholes 2008 via oceanexplorer.noaa.gov.

It took a few billion years from the evolution of the first photosynthetic cyanobacteria to the time when there was enough oxygen in the atmosphere to support animal life like us. Why did it take so long? NPR interviews scientists investigating purple microbial mats in Lake Huron.

Citing this post: Urbano, L., 2016. Carbon in the Ground and Free Oxygen in the Air, Retrieved February 23rd, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# MolView: Website for Drawing Molecules

#### September 5, 2016

MolView is a great site for drawing molecules and rendering them in 3D.

Screen capture from MolView.

Citing this post: Urbano, L., 2016. MolView: Website for Drawing Molecules, Retrieved February 23rd, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# John Snow: How to be a Scientist

#### May 22, 2016

These three excellent, short videos on John Snow’s life and work on cholera do a nice job of describing what makes for good science–careful observation; good notes; creative analysis of data, etc. They should make a good “spark your imagination” introduction to biological science.

They also have an excellent explanation of all the ‘lies’ and liberties they took in the making of the video.

Citing this post: Urbano, L., 2016. John Snow: How to be a Scientist, Retrieved February 23rd, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.