Day 7: Convection
by Youry Aglyamov
I don’t really like blog posts filled with physics equations. I tend to skim over them while reading—I only get engrossed by quantitative physics when I’m actually doing it. Unfortunately, quantitative physics was most of what we did today, so that’s what I’ll be talking about.
So. We’re done with Mars, moving on to Jupiter while basically skipping over Venus & Mercury. We started with a discussion of the giant planets’ magnetic fields—Jupiter’s is tilted and off-center, whereas Saturn’s is in alignment with its axis and on-center. (Earth’s is centered but tilted.) Uranus and Neptune are completely weird, but it seems scientists know why, as we’ll cover—next week.
(Uranus is my favorite planet, insofar as I have one. When I was younger, it used to be Pluto, but, well…. That said, Jupiter’s pretty awesome too.)
Jupiter and Saturn’s fields haven’t flipped in the time we’ve been observing them from Earth, that is, since 1970. We can’t observe the fields of Uranus and Neptune, because they’re too far away to see the fields from Earth and no one has sent anything there since Voyager 2. If you wanted to measure whether Jupiter’s flipped in the past, the least impossible option would be to sample Io’s lava flows and date them, comparing their magnetic signatures and doing calculations to see how Jupiter’s magnetic field would affect that. Actually, scratch that: the least impossible option would be to wait a few million years and, if Jupiter still hasn’t flipped, conclude it never does. We assume all layers are evenly mixed in all chemicals because convection – except clouds and phase changes lead to weirdness, so this doesn’t apply to water, ammonia, and other cloud-forming compounds. One would think that lots of compounds form clouds somewhere in Jupiter’s enormous molecular hydrogen layer. I mean, if 2 Mbar won’t cause something to condense, what will? (Although it’s at 6500 K, so….)
Then we split into four groups. There were three graduate student geophysicists in the room, plus myself, a freshman geophysics major. (I haven’t even taken a geophysics course yet!) So when Professor Brown said that this assignment would be easy because the geophysicists had already learned it, I suddenly got worried.
At least we weren’t doing particularly esoteric stuff—convection, ignoring pressure differentials and variability in the acceleration of gravity. If we raise a cubic parcel of fluid x3 up a distance z in a location where the temperature gradient is dT/dz, the buoyant force upward is x3 g∆ρ, where ∆ρ is the difference in density between the parcel and the outside fluid in its location. Here our group split—I raced ahead to find the expression for ∆ρ, made a mistake, realized it, made another mistake, got the wrong answer, retraced my steps, and ultimately got F=gαρ1 x3 ∆T, where F is force, g is the acceleration due to gravity, α is the thermal expansion coefficient, ρ1 is density (the density of the outside fluid, as I got it), and ∆T is the difference in temperature. I tried to work with the rest of the group, but in my excitement and repeated mistakes lost track of what they were doing completely.
They presented at the board as I finished my own derivation. They got the answer slightly wrong because they calculated ρ1=ρ0/α∆T instead of ρ1=ρ0/(1+α∆T), where ρ0 is the density of the parcel, but the essentials were correct. Unfortunately, because we moved so fast, Professor Brown simply wrote down the correct answer on the board (it was F=gαρx3 ∆T, but which density wasn’t specified) and gave us a new problem. This time it was the question of the viscous drag force on the particle from a medium with viscosity & (it wasn’t actually an ampersand, he just drew a squiggle on the board), if the particle has velocity v and area A. The answer, by dimensional analysis, is sqrt(A). Why is A under the radical? Because the units require it. Because of the rushed pace, however, we took a while to find the mistake in the group’s calculations for the first problem; eventually Professor Brown pointed it out, but by that point the complete chaos of the end of class was upon us.
As those last five minutes began, we first talked about viscosity, then wrote down terminal velocity for the parcel of air falling upward: (x2 gαρ∆T)/&. This terminal velocity, you will note, increases as the particle falls, so the only reason it doesn’t keep falling upward forever is that it cools. Specifically, it cools according to its thermal conductivity: dQ/dt=κ∆T/x, as we were told at the end of class, where κ is thermal conductivity. And somehow, all of those constants can be combined to get the Rayleigh number, which tells us if the fluid column is stable. Before we could make sense of this, or for that matter shown it in a quantitative manner, we ran out of time and class was dismissed.
This week we leave Mars behind and move on to talk about giant planets in the Solar System. The giant planets generally include Jupiter, Saturn, Uranus, and Neptune, and they are composed of mostly gases. For example, Jupiter has a thick atmosphere of hydrogen and helium; inside the atmosphere it has a layer of metallic hydrogen, surrounding a core of molten rock. With such different layers, an important question is whether these layers are well-mixed, and if yes, which mechanism(s) actually does the job.
A major mechanism to mix things on the planetary scale is convection, and it is also responsible for the layer mixing here. We can think about a simplest scenario of convection by considering a cube of gas with volume d3 on a hot surface. Beyond the surface there is a temperature gradient throughout a height of H, with the top being the coldest. Now due to buoyant force, the cube of gas will rise so that it is hotter than its surroundings. We can imagine that this cube will keep rising until it cools down or meets resistance from other forces.
The first question to ask is, how much is this buoyant force? We know as temperature rises, the volume of an object will expand because the molecules move faster to occupy more space, given a constant pressure. We introduce a coefficient of thermal expansion here as: α = (1/V)(dV/dT), which indicates a fractional change in the volume when temperature changes. For a temperature change of ΔT, suppose the volume change is ΔV, then ΔV = α * V * ΔT.
So the new total volume is V + ΔV = V * (1 + α ΔT).
The original density of the cubic is ρ = mass/V, so the new density when temperature changes by ΔT is ρ’ = mass / V / (1 + α ΔT) = ρ / (1 + α ΔT).
The buoyant force comes from the density difference between the hotter cube and the colder environment. So Fb = ( ρ – ρ’)*d3*g is the buoyant force.
Now we think about what could resist the cube from rising. First there is the viscosity from the surrounding fluid. To represent viscosity, we think about what physical quantities would matter: how viscous the fluid is, how fast the cube moves, and how big the cube is. After dimension analysis, we come up with the viscous force as: Fμ = μ*v*d.
Note that d is the typical length scale in this case, instead of d2 as the contact area of that cube with its environment.
If we equate the buoyant force with the viscous force, we can find the equilibrium velocity of the rising cube.
In order to stop rising, the cube can also cool down by conduction. The rate of cooling down is again derived by collecting physical terms that matter and dimensional analysis: dT/dt depends on k, d, and ΔT, where k is the conductivity coefficient.
There are criteria for convection to be possible, and we will discuss these in details later.
At the beginning of class, we discussed a few questions prompted by the video lectures. One major topic revolved around the “dry spot” Professor Brown talked about in relation to the Galileo probe observing anomalously low concentrations of water on Jupiter. Some members of the class were wondering why a “dry spot” would occur. We discussed possible reasons for this phenomenon and settled on the possibility that the pressure and temperature conditions on Jupiter were close to a phase transition for water. Thus, water vapor in the atmosphere of Jupiter could condense into clouds of liquid and solid water, leaving a relatively dry area that the Galileo probe happened to fall into. The analogy put forward goes as follows: a probe could approach Earth and happen to fall into the middle of a major desert, like the Sahara or the Gobi. Observing Earth would clearly show that large oceans existed, however, the probe simply happened to fall into an arid region.
Next the class made some attempts at deriving the equations governing convection. Qualitatively, the approach consisted of considering a cube of infinitesimal volume d3 floating in liquid with a constant temperature gradient cooling as you move upwards. We assumed constant pressure for simplicity and first only considered the buoyant force on the cube. If the tiny cube moves up a small amount, its density will be less than its surroundings, because the cube remains hotter than the liquid it is moving through. As anyone who has played in a bathtub can tell you, less dense objects float. Thus the cube will continue to move upwards, conceivably until it hits the surface of the liquid.
But we all know this isn’t how convection works, so there must be other parameters our simple model isn’t considering. Viscosity is one. Imagine trying to swim through a lake of chocolate syrup. The “resisting” force the liquid is applying on you is viscosity. Into our model we added a viscosity component, whereby the liquid would hinder the motion of the cube. A little bit of dimensional analysis showed us that viscous force increases with velocity, viscosity of the liquid, and the square root of the cross sectional area.
Finally, we added a conduction component, necessary to allow our volume sink after some period of rising and complete the “cycle.” As the cube rises, it exchanges heat with its surroundings through conduction until finally they are the same temperature. At this equilibrium point, the cube is the same density as the liquid it is floating in and begins to turn around. That’s all we had time for in class, but the adventure will continue in office hours this week!
Day 8: Stupid Gravity Tricks
We gathered into 4 groups of 3 to discuss four topics: heat flow, the Love number, spherical harmonics and gravity anomalies on the Moon.
A. Heat Flow
How do we know the Earth is hotter inside?
(1) The Earth has a magnetic field, which requires molten magnetic material.
(2) Volcanoes exist.
(3) Seismic tomography can provide us insight into density, sound velocities and phase transitions, which correspond to specific pressures and temperatures.
(4) The slope of temperature vs. depth even at shallow depths tells us that T increases as we dig deeper.
We discussed Lord Kelvin's under-approximation of Earth's age through his analysis of heat flow and today's current surface temperature by sketching a temperature profile as a function of depth and time. Measuring dT/dz in mines and matching the slope to one of the temperature profiles can give us a corresponding age of the Earth. However, this does not take into account the heat released by isotopic decay, which is the reason Earth is much older than Lord Kelvin's prediction.
B. Love Number
How rigid is a planet? We can assess a planet's rigidity by looking at how it deforms due to rotation, a moon's gravitational pull and the sun's gravitational pull. A useful quantity is called the Love number, which can be defined as the ratio of the induced potential to the the disturbed (rotational/tidal) potential. A perfectly rigid planet would have no induced potential and so would have a Love number of 0, whereas a perfectly fluid one would have a value ~1.5. We can determine a planet's Love number by gravitational potential variations with time and using a laser altimeter. Some example values are: Earth~0.3, Venus~0.3, Mars~0.15, Titan~0.5, Mercury~0.45.
C. Spherical Harmonics
Enceladus is a small icy moon of Saturn (it can fit between Los Angeles and San Francisco!). It has geysers on the south pole as a result of “yanking” by Saturn, suggesting the presence of water beneath the ice. Using a spherical harmonic representation of the gravitational potential and solving for the C_30 coefficient that describes the mass asymmetry between the north and south hemisphere, it was possible to determine that there was in fact a mass anomaly in the southern portion of Enceladus. This was exciting confirmation that an ocean was located at and only at the south pole.
D. Gravity Anomalies on the Moon
Two satellites orbit the moon together and measure the gravity anomalies through the change in the distance between the them. One can imagine two satellites (A and B) passing over a mountain. First, in a non-anomalous region, the satellites' distance is constant. When satellite A passes over a mountain, the extra gravity due to the extra mass of the mountain pulls slightly at satellite A, increasing the distance between satellite A and B and making satellite A move faster.
Interesting results of this mission:
(1) The near side of the moon had a much higher Bouguer anomaly (a gravity anomaly without the effect of topography) than the far side.
(2) The density of the crust was 12% lower than we previously thought.
(3) The Aitken basin on the south pole had very dense crust, likely as a result of the impact that formed it.
(4) There are many linear features corresponding to dense structures. They appear similar to dikes on Earth and they predate all the craters. One hypothesis (Andrews-Hanna et al., 2013) is that as the Moon accreted, the inside was temporarily cooler and so when it heated up, the surface expanded. This expansion would result in fracturing and subsequent dike formation as dense material filled in those fractures.
by Alice Michel
At the core of today’s class was gravity! It was “Gravity Tricks Day,” science camp-style. Basically, today’s class took on a slightly different format, in which each of four groups rotated between four stations at which the TAs and professor each discussed a specific bit to the gravity story.
Station 1: Enceladus: Saturn’s Watery Moon?
Often in planetary science, the search for liquid water is a driving force. This story is one such example. Enceladus is a ~500 km diameter moon of Saturn. Images have been taken of Enceladus that show plumes coming out of the south pole. Were they water? Gravity to the answer! The spacecraft Cassini collected data on the time it took to circumnavigate the planet and the distance, so we know the velocity, which can be directly related to the gravitational potential, V, of a planet since the change in velocity over time is acceleration, which equals –GM/r2, from which we can find V in terms of r and two angles. Then we looked at a crazy equation with spherical harmonics in it. Really it’s just the equation for a spherically uniform body or point source, V = GM/r, plus spherical harmonics, which are just basis vectors, from linear algebra, in 3D. It still sounds kind of scary, but the gist of it is that, given V, which we have since we have v, we can solve for some polynomial coefficients in the spherical harmonics part of the equation. These polynomials tell us about the mass distribution of the planet, with one giving us the difference in mass distribution between the equator and the poles and the other between the two hemispheres of a planet. When scientists did this using the Cassini data, they found that there is a huge mass anomaly in the south of Enceladus. It’s 10 km thick and buried in 30–40 km of ice. What does this mean? More evidence for liquid water!
Station 2: Our Moon and the GRAIL Mission
Scientists are at work doing gravity tricks with our own moon, too. The GRAIL mission of last year used topographical variations on the moon to map gravity anomalies using the fact that two spacecraft a given distance apart would become more spread out if one experienced different gravity. They found that the acceleration due to gravity on the moon varied by up to 1000 mgals (mm/s2). Then they subtracted the topographical contribution to gravity, leaving only the gravity anomalies due to the subsurface. This information is super important because it showed that, (1) as expected, it’s relatively constant, (2), knowing about gravity anomalies is important when planning missions to the moon that could get off-course, and (3), we know the far side crust is thicker due to stronger gravity. The most interesting thing we learned at this station is that there are these linear high density anomalies, which look a lot like dikes on Earth. This is cool because it means that magma leaked into cracks in the lunar subsurface at one point, which means that the moon was cold inside and had a hot shell when it formed. This supports the protolunar disk hypothesis for the formation of the moon! Another thing that I thought was interesting at this station is that this method of mapping gravity is how we map the seasonal and yearly size of the polar caps on our own Earth.
Station 3: The Interior of Earth AKA Thinking Like Crazy Old Dead Guys
It’s pretty clear to us that the interior of the Earth is hot. There’s no doubt about it actually. But how did people know this hundreds of years ago? First off, there are volcanoes! But people actually had an even better grasp than that. Lord Kelvin used the temperature gradient of deep caves to estimate the age of the Earth a long time ago. His final answer? Ninety-four million years. (Hey, it was better than the 6,000 years Biblical estimate.) Okay, so how did that work? At this station, we tried to work through his thought process by drawing curves showing temperature change with depth (up to 6000 km) and time assuming Earth is entirely granite (which has a melting temperature of 1600 K), the surface temperature of Earth is 300 K, there is no convection, and no radioactivity. (Yes, drastic assumptions, we know.) My group’s final curves looked almost exponential, but with end points on the 1600 K boundary line that creep along to 6000 km deep as time increases. So now all we have to do is find a deep cave to calculate the temperature gradient and match it to our curves, and we’ll be just like Lord Kelvin.
Station 4: Love Number
In geophysics, "love numbers" measure the stretchiness of a planet. This is directly related to what the planet is made out of. Rigid planets are going to be less stretchy and thus have a lower love number. Mathematically, the fluid love number, h, can be found by multiplying the radius change caused by planet rotation and/or tides, dr, by acceleration due to gravity, g, and dividing by the stretch, V, due to planet rotation and/or tides. Then h = 1 + k where k is the induced V divided by V from before. In other words, you can figure out the stretchiness of a planet without landing on it - so you can figure out what it is made of without landing on it! k = 0 represents a perfectly rigid planet whereas k = 1.5 represents a fluid one. Some quick samples: k of the Earth = 0.3, Venus = 0.3, Mercury = 0.45 (very dense with a huge liquid iron core), and Titan = 0.6 (interesting because this supports the notion that there is liquid water beneath the icy surface, though there is then more ice beneath that, which detracts from the life theory a bit).
With that, class time was up. All in all, I found the “stations” format to be really nice, and I hope we do it again. It worked well at packing in lots of different pieces of information while still getting us to individually consider the implications.
Stupid Gravity Tricks: The theme of today’s class, verbatim from our esteemed Mike Brown.
As such, we spent the day cycling between the TA’s and Mike to discuss various gravitational concepts and practices implemented by scientists to determine the structure of planetary bodies.
My group first sat down to learn the basics of what a love number is, and its effect on a planetary extension. If we take a planet and assign it some density variations (which, in practice, we could figure out with spacecraft), we would observe deformations in the planet due to (a) rotational effects and (b) tidal effects. What we want to specifically look at are the tidal effects, as these are what really allow us to differentiate planetary composition from one planet to another. On earth, we observe body tides as an elongation towards either the sun or the moon. This elongation, δr, can be expressed as approximately h(V/g), V is the gravitational potential, g is the gravitational constant, and h is the fluid love number. A second type of love number, the tidal love number, is represented as k and is approximated by Vinduced/Vrotational,tidal. By calculating the love number, or, observing the change in flex of a planetary body, we can learn more about its structure and interior. By comparing the love numbers of different bodies, we can relatively say whether a body is more “elastic” or “fluid.” (At this point, we were given a list of different planetary love numbers. If you are interested in knowing the numbers, leave a comment below.)
Next, we explored Saturn’s moon, Enceladus, famous for geysers that burst from its southern pole. It is quite small, able to fit between Los Angeles and San Francisco, yet has extremely active geysers that spout from its subsurface ocean. Enceladus has been the recent topic of conversation among planetary scientists for aforementioned subsurface ocean and its potential to host life. But this is Stupid Gravity Tricks day and not Paleobiology day. So let’s get back to the gravity of the situation. As the orbiter Cassini flies by Enceladus, we know the velocity at which the spacecraft is traveling: as the craft is constantly sending back radio waves, from Doppler shifts we can determine velocity changes. If we then take the derivative of the velocity, dv/dt, we then know the acceleration. The acceleration, ag, is equal to –GM/r2 which is equal to –ΔV, the gravitational potential. V is a function of r, θ, and ϕ, (spherical coordinates) meaning that it is represented with spherical harmonics. To breakdown this function and visualize it on a more tangible scale, we turn towards an expansion of the gravitational potential expression. Just we can apply a Fourier transform to a wave packet to revert it into its elementary wave components, we can break down gravitational effects from complex, non-spherical, three-dimensional surfaces.
The expression for gravitational potential may look (or may not, depending on your comfort level with physics) daunting, however the concept is extremely basic. It is expressed in spherical coordinates, which is equal to GM/R times something complicated. Just looking at GM/R, this section of the expansion represents the gravitational potential due to a perfectly spherical surface. Everything after this point is just additional complexities to the surface structure. Clm and Slm are weight coefficients that essentially tell us what is inside the planetary body due to the gravitational effects. When we apply this to Enceladus, we are able to determine an asymmetry between the northern and southern poles, which tells us the presence of a subsurface ocean at the South Pole, but not the North.
The GRAIL mission sent two spacecraft that simultaneously orbited the moon. GRAIL 1 was followed by GRAIL 2, and by measuring slight differences in data collected by the two, an in-depth construction of surface, bulk density (AKA things like mountains and other topography) could be created. On a simplified scale, as GRAIL 1 approaches a large mountain, the slight gravitational effect would pull the spacecraft slightly faster, and away from GRAIL 2. Then GRAIL 2 would begin to feel the same effect while GRAIL 1, having passed the mountain, slows down. Then as they both pass the mountain, the original equilibrium is achieved and they continue to fly onwards. We can measure the changing distances between the two spacecraft and determine the gravitational effect of whatever surface structure they passed. There is a lot of very compelling information that can be drawn from the data that is best represented in the image-recreations. If this topic interests you, I encourage you to explore it further by reading the NASA article here.
In this section led by Mike Brown, we were asked the question: How do we know the inside of the earth is hot? We have evidence from “stuff coming out of the inside of the earth” such as volcanic activity releasing extremely hot material. From studies of seismic waves (p-waves and s-waves), we can determine that the earth has a liquid core. Additionally, when going down a mine, the temperature gets warmer. When Lord Kelvin attempted to determine the age of the earth, he used this information and some rough approximations in order to create a temperature profile of the earth throughout history. If we can say that the cooling of the earth occurs though a constant flux F ∝ dT/dz, we can assign thermal trends of temperature vs. depth for the entirety between an initial condition, with all of the earth, except for the surface, at the hottest temperature until the entire earth is at the coldest temperature. From this linear progression, he examined the temperature gradient towards the surface of the earth and compared it with what he could observe by going down into a mine. From this, he calculated that the earth was 94 million years old. Which is quite off, but still a valiant effort and a good exercise in applications of thermal analysis. (What did this have to do with gravity?)