"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "skip"
}
},
"source": [
"> Hannes Grobe/AWI, http://commons.wikimedia.org/wiki/File:Iceage_north-intergl_glac_hg.png"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"\n",
"### Pacing of ice ages: evidence from ocean sediments"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The figure below shows a global record of **oxygen isotopes** recorded in the shells of marine organisms. This record tells us primarily about variations in **global ice volume** -- because the net evaporation of water from the oceans to build up the ice sheets leaves the oceans enriched in heavier isotopes.\n",
"\n",
"The x axis is plotted in **Thousands of years before present** (present-day is at zero on the left).\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"> Lisiecki, L. E. and Raymo, M. E. (2005). A Pliocene-Pleistocene stack of 57 globally distributed benthic δ18O records. Paleoceanog., 20."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"The ice ages (times of extensive glaciation and high ocean $\\delta^{18}$O) do not seem to be random fluctations. They have come and gone (approximately) periodically, somewhat like the seasons."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "-"
}
},
"source": [
"**Spectral analysis** of such records reveals peaks at some special frequencies:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"> Imbrie, J. and Imbrie, K. P. (1986). Ice Ages: Solving the Mystery. Harvard University Press, Cambridge, Massachusetts."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"The peaks noted on this figure are special because they correspond to frequencies of variations in Earth's orbital parameters, as we will see.\n",
"\n",
"These kind of results became available in the 1970’s for the first time, because ocean sediment cores allowed a sufficiently detailed look into the past to use time series analysis methods on them, e.g. to compute spectra.\n",
"\n",
"The presence of peaks in the spectrum at orbital frequencies was seen as convincing evidence that the so-called **astronomical theory of the ice ages** was (at least partially) correct.\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"____________\n",
"\n",
"\n",
"## 2. Introducing the astronomical theory of the ice ages\n",
"____________\n",
"\n",
"The **Astronomical Theory** of climate and the ice ages looks to the **regular, predictable variations in the Earth's orbit around the Sun** as the driving force for the growth and melt of the great ice sheets. Such theories have been discussed since long before there was any evidence about the timing of past glaciations."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"Last time we saw that insolation is NOT perfectly symmetrically distributed between the two hemispheres and seasons. \n",
"\n",
"To refresh our memory, let's use \n",
"```\n",
"climlab.solar.insolation.daily_insolation()\n",
"```\n",
"to compare the maximum insolation received at the North Pole (at its summer solstice) and the South Pole (at its summer solstice)."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Daily average insolation at summer solstice:\n",
"North Pole: 525.31 W/m2.\n",
"South Pole: 562.03 W/m2.\n"
]
}
],
"source": [
"from climlab.solar.insolation import daily_insolation\n",
"days = np.linspace(0, const.days_per_year, 365)\n",
"Qnorth = daily_insolation(90,days)\n",
"Qsouth = daily_insolation(-90,days)\n",
"print( 'Daily average insolation at summer solstice:')\n",
"print( 'North Pole: %0.2f W/m2.' %np.max(Qnorth))\n",
"print( 'South Pole: %0.2f W/m2.' %np.max(Qsouth))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"These asymmetries arise because of the detailed shape of the orbit of the Earth around the Sun and the tilt of the Earth's axis of rotation.\n",
"\n",
"As these orbitals details change over time, there are significant changes in the distribution of sunlight over the seasons and latitudes."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### The Milankovitch hypothesis\n",
"\n",
"Version of the astronomical theory have been debated for at least 150 years.\n",
"\n",
"The most popular flavor has been the so-called **Milankovitch hypothesis**:\n",
"\n",
"> Ice sheets grow during periods of *weak summer insolation* in the Northern high latitudes."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"The idea is that for an ice sheet to grow, seasonal snow must survive through the summer. Milankovitch therefore focussed on the factors determining the climatic conditions during **summer**."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"____________\n",
"\n",
"\n",
"## 3. Ellipses and orbits\n",
"____________\n",
"\n",
"First, watch this neat animation from [Peter Huybers](http://www.people.fas.harvard.edu/~phuybers) (Harvard University):\n",
"\n",
"http://www.people.fas.harvard.edu/~phuybers/Inso/Orbit.mv4\n",
"\n",
"Watch carefully and note the three ways that the orbit is varying simultaneously."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"From Professor Huybers' web page:\n",
">A movie depicting Earth's changing orbit over the last 100Ky. The orientation is such that spring equinox (indicated by a vertical bar) is directly to the front with the sun behind it. Northern Hemisphere summer is to our right, and Northern Hemisphere winter is to the left. The apsidal (dashed) line connects perihelion (Earth's closest approach to the sun) to aphelion (the point when Earth is furthest from the sun). The rotaion of the apsidal line occurs because of the precession of the equinoxes and has a roughly twenty-two thousand year period. The semi-circle around the Earth indicates the location of the equator and the straight line is the polar axis. Obliquity is defined as the angle beetween the orbital and equatorial planes. The variations in Earth's obliquity and the eccentricity of Earth's orbit have both been increased in magnitude by a factor of ten. Also, the Earth's angular velocity has been decreased by a factor of five thousand. Note that Earth's angular velocity is slowest at aphelion and fastest at perihelion. \n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"The Earth’s orbit around the Sun traces out an **ellipse**, with the Sun at one focal point."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"> Imbrie, J. and Imbrie, K. P. (1986). Ice Ages: Solving the Mystery. Harvard University Press, Cambridge, Massachusetts."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### How to draw an ellipse\n",
"\n",
"1. Take any two points on a plane\n",
"2. Attach the two ends of a piece of string to the two points. \n",
"3. Pull the loose string out as far as it will go in any direction, and place a pencil mark at that point.\n",
"4. Do the same for every possible direction.\n",
"5. Congratulations, you have just drawn a perfect ellipse. The two points are called **foci** or focal points."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"Keep this in mind, and you will always understand the **mathematical definition of an ellipse**:\n",
"\n",
"> An ellipse is a curve that is the locus of all points in the plane the sum of whose distances from two fixed points (the foci) is a positive constant.\n",
"\n",
"In our case, the positive constant is the total length of the string."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Perihelion and Aphelion\n",
"\n",
"The point in the orbit that is **closest to the sun** is called **Perihelion**.\n",
"The farthest point is called **Aphelion**.\n",
"\n",
"Distances (present-day): \n",
"\n",
"- Perihelion, $ d_p = 1.47 \\times 10^{11}$ m\n",
"- Aphelion, $ d_a = 1.52 \\times 10^{11}$ m"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Eccentricity\n",
"\n",
"The eccentricity of the orbit is defined as\n",
"$$ e = \\frac{d_a-d_p}{d_a+d_p} $$\n",
"\n",
"So for present-day values, $e = 0.017 = 1.7\\%$\n",
"\n",
"Earth’s orbit is nearly circular, but not quite! \n",
"\n",
"(What value of $e$ would a purely circular orbit have?)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"As the Earth travels around its orbit, the **distance to the sun varies**. The energy flux (W m$^{-2}$) is larger when the Earth is closer to the sun (i.e. near perihelion).\n",
"\n",
"At present, perihelion occurs on January 3. This is very close to the Northern Hemisphere winter solstice (Dec. 21)."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"**The Earth actually receives MORE total sunlight during Northern Winter than during Northern Summer.**\n",
"\n",
"It is thus critical to understand the relative timing of our seasons (which are determined by the axial tilt or obliquity) and the perehelion."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Obliquity\n",
"\n",
"The obliquity $\\Phi$ is the **tilt of the Earth’s axis of rotation with respect to a line normal to the plane of the Earth’s orbit around the Sun.** \n",
"\n",
"Currently $\\Phi = 23.5^\\circ$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"Obliquity is the fundamental reason we have seasons, and would have seasons even with a perfectly circular orbit ($e=0$)\n",
"\n",
"Higher obliquity means:\n",
"\n",
"- more summertime insolation at the poles\n",
"- less wintertime insolation in mid-latitudes"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Longitude of perihelion and precession of the equinoxes\n",
"\n",
"The **longitude of perihelion** is defined as the angle $\\Lambda$ between the Earth-Sun line at vernal equinox and the line from the Sun to perihelion (see sketch).\n",
"\n",
"The current value is $\\Lambda = 281^\\circ$ (perihelion on January 3, shortly after NH winter solstice).\n",
"\n",
"We call the gradual change over time of the longitude of perihelion the **precession of the equinoxes** (or just precession). It is the **gradual change in the time of year at which the Earth is closest to the Sun**."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"#### Question\n",
"\n",
"Can there be any precession for a planet with a *perfectly circular* orbit (zero eccentricity)?"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"It is important to understand that eccentricity modulates the precession. *Highly eccentric orbits lead to larger differences in the seasonal distribution of insolation.* \n",
"\n",
"We quantify this with the **precessional parameter**\n",
"\n",
"$$ e \\sin \\Lambda $$\n",
"\n",
"Large positive precessional parameter = Excess insolation during summer in the northern hemisphere."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### The three orbital parameters\n",
"\n",
"We have just identified three parameters that control the seasonal and latitudinal distribution of insolation: $e, \\Lambda, \\Phi$\n",
"\n",
"All three vary in predictable ways over time. They have been calculated very accurately from astronomical considerations (basically the gravity of the Earth, Sun, moon, and other solar system objects).\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"____________\n",
"\n",
"\n",
"## 4. Past orbital variations\n",
"____________\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"There are tools in `climlab` to look up orbital parameters for Earth over the last 5 million years.\n",
"\n",
"We will use the package\n",
"```\n",
"climlab.solar.orbital\n",
"```"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Data retrieved from http://thredds.atmos.albany.edu:8080/thredds/fileServer/CLIMLAB/orbital/orbit91 and saved locally.\n"
]
}
],
"source": [
"from climlab.solar.orbital import OrbitalTable"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"| Software | Version |
|---|---|
| Python | 3.7.3 64bit [Clang 4.0.1 (tags/RELEASE_401/final)] |
| IPython | 7.6.0 |
| OS | Darwin 17.7.0 x86_64 i386 64bit |
| numpy | 1.16.4 |
| matplotlib | 3.1.1 |
| climlab | 0.7.5 |
| Wed Jul 03 14:53:31 2019 EDT | |