Final review for Climate and Climate Change, Spring 2025#
Final exam logistics#
Wednesday, May 21st, 1-3:15pm, MR829
100% closed to all resources: notes, books, internet, classmates
unlabeled list of key equations will be provided
Mostly multiple choice; one or more short answer
Comprehensive over whole semester, but moderately weighted toward material after the midterm
Roughly same length as midterm
Introductory lecture (2025/2/5)#
inverse square law: dependence of stellar radiation, in \(\text{W m}^{-2}\), on a star’s luminosity and orbital distance
This is the starting point for Earth’s climate: how much energy does Earth get from the sun?
Basic properties of Earth’s climate (various lectures)#
Atmospheric structure#
Vertical structure: troposphere is lowest ~10-15 km, where ~all weather happens, and above it is the stratosphere, separated by the tropopause
top of atmosphere is where planetary energy balance is determined: for balance, must have the sum total of all radiative fluxes in equal to the sum total of all radiative fluxes out
Horizontal structure: tropics vs. extratropics
Tropics further subdivided into deep tropics (~15\(^\circ\)S-15\(^\circ\)N) and subtropics (~15\(^\circ\)S-30\(^\circ\)S and ~15\(^\circ\)N-30\(^\circ\)N)
Extratropics further subdivided into midlatitudes (~30\(^\circ\)S-60\(^\circ\)S and ~30\(^\circ\)N-60\(^\circ\)N) and high latitudes (~60\(^\circ\)S-90\(^\circ\)S and ~60\(^\circ\)N-90\(^\circ\)N)
Moisture#
specific humidity (\(q\)): in a given volume, (kg water vapor) / (kg air)
saturation: when air is at exact temperature and amount of moisture set by Clausius-Clapeyron such that any more water vapor (or any cooling) would cause the water vapor to start condensing out into liquid or ice water
saturation specific humidity: value of \(q\) at saturation
relative humidity: (actual specific humidity) / (saturation specific humidity)
Greenhouse effect (2025/2/19)#
Fundamentals#
blackbody radiation and Stefan-Boltzmann law: \(E=\sigma T^4\)
two-band approximation: shortwave radiation emitted by sun (abbreviated SW), vs. longwave radiation emitted by Earth (abbreviated LW)
transmission (photon passes through) vs. scattering (photon reflected) vs. absorption (photon absorbed and ceases to exist)
Planetary energy balance#
insolation: incoming solar radiation
planetary albedo (\(\alpha_p\)): fraction of insolation reflected back to space; thus the fraction \(1-\alpha_p\) is absorbed by the climate system
outgoing longwave radiation (OLR) and corresponding emission temperature: \(T_E=(OLR/\sigma)^{1/4}\)
All insolation passes through a disk w/ Earth’s radius, so area \(\pi a^2\) where \(a\) is Earth’s radius, Whereas OLR is emitted over surface area of Earth, so area \(4\pi a^2\). This difference in the areas between the SW and LW is why the 1/4 factor appears in the planetary energy balance temperature
Greenhouse effect#
greenhouse effect: surface is warmer than the emission temperature due to absorption and re-emission of LW radiation by the atmosphere
This absoprtion and reemission is done by greenhouse gas molecules
Key ones are CO2 and water vapor.
Water vapor is arguably single most important greenhouse gas
But its concentration is set by temperature via Clausius Clapeyron: for fixed relative humidity, warmer = more water vapor = more greenhouse
So better thought of as a feedback than a forcing
Whereas CO2, humans are emitting more into the atmosphere, causing warming and associated radiative feedbacks and other climate system responses
1-layer greenhouse model#
If there was no greenhouse effect, the surface temperature and emission temperature would be identical: the photons emitted by the surface would all escape to space.
That’s a good description of the moon, which basically has no atmosphere and thus no greenhouse effect. But it’s a bad description of Earth.
The simplest analytical framework that we can use to represent the greenhouse effect is the one-layer greenhouse model.
Emissivity and absorptivity#
Earth’s atmosphere is not a perfect blackbody. We quantify this using emissivity, \(\epsilon\): given a substance’s actual emitted radiative energy in \(\text{W m}^{-2}\), the fraction \(0<\epsilon\leq1\) this is of the true blackbody radiation that substance would emit, given its actual temperature.
By Kirchhoff’s law, the emissivity and absorptivity are identical. So if, for example, suppose the atmosphere’s emissivity for longwave radiation was measured to be 74%, then it must also be the case that it absorbs 74% of the longwave radiation that it receives from the surface below. The remaining \(1-\epsilon\), which in this example is 26%, passes through without being absorbed.
Emission height and lapse rate#
For unperturbed climate (i.e. unchanged CO2 amount)#
emission height: determined directly by the atmospheric composition, i.e. how many molecules there are of the various greenhouse gases (GHGs)
For Earth combining all the GHGs, this is ~5-6 km
lapse rate: change in temperature with height, with sign flipped.
Within the troposphere, the lowest atmospheric layer, the lapse rate on average is ~6 K/km
Under increased CO2: (emission height rises) + (~fixed lapse rate) = (surface warming)#
More CO2 = vertically higher emission height = colder temperature = less blackbody radiation = less OLR
Must increase temperature at this new emission height to the original one
If lapse rate fixed, this requires shifting the whole temperature profile toward higher values
Results in warming at the surface (and the whole troposphere)
Spectral nature of radiation#
Bulk SW and LW categories are in fact averages over large bands of wavelengths: ~0.1-2 microns for SW, ~5-80 microns for LW
absorption band: greenhouse gases are those gases that have absorption bands within the overall LW part of the spectrum
E.g. CO2 15 micron band: absorbs radiation emitted at and near 15 microns
The location and strength of the absorption bands is determined by quantum mechanics and each molecule’s structure
Radiative forcing#
anthropogenic = caused by humans
radiative forcing: Immediate change in TOA net radiative flux after some change has been imposed within the climate
E.g. suppose CO2 was instantly doubled, the best estimate is that this would cause a ~4 W/m^2 imbalance at TOA, meaning the climate system is receiving 4 W/m^2 more than it is emitting to space
A nonzero radiative forcing means causes the climate system to respond in order to reach a new equilbrium
Temperature (2025/2/26, 3/5)#
Climate sensitivity#
climate sensitivity: global-mean surface air temperature change in response to a given radiative forcing, usually a CO2 doubling.
equilibrium climate sensitivity vs. transient climate sensitivity: ~1000s of years for equilibrium vs. ~1st century after a radiative forcing is introduced for transient.
Why? Right now, ~90% of extra energy being absorbed in response to the increased CO2 is being absorbed by the ocean, and virtually all of that in a thin layer near the surface (say the first km, or even less). The deep ocean (from ~1 km to bottom) is huge heat reservoir, but takes up heat from near surface very slowly, so it takes 1000s of years for it to fully equilibrate
Thus, equilibrium warming can be substantially greater than the transient warming
Radiative feedbacks#
Others#
polar amplification and its 3 main contributing factors: Planck, lapse rate, and surface albedo feedbacks
hiatus periods and their underlying cause: long-term trend ((a.k.a. secular trend) with natural variability overlaid
stratospheric cooling in response to increased CO2, at the same time as troposphere underneath warms
Clouds (2025/3/6, 2025/3/12)#
Basics#
3 key types in terms of vertical structure: low cloud, high cloud, cumulonimbus
Liquid vs. ice: liquid if air is ~above freezing, ice if much below freezing, can be “mixed-phase” with both ice and liquid present if temperature is intermediate (between 0 and -40 Celsius)
Cloud radiative effect#
cloud radiative effect (CRE): all-sky TOA radiative flux minus clear-sky TOA radiative flux, in Watts per meter squared
Can consider SW and LW separately: SW CRE and LW CRE, or add them to get the net CRE = SW CRE + LW CRE
SW CRE#
All clouds have high albedo: more SW reflected back to space compared to if that cloud was absent
So SW CRE ~always negative: clouds cause less SW to be absorbed by the climate system
(Quantitative value depends on the cloud’s own albedo relative to the underlying surface albedo)
LW CRE:#
Clouds are ~blackbodies: LW emission varies with \(T^4\)
Being H2O, clouds strongly absorb LW
Combined: they absorb LW emitted below them, then re-emit it at their local temperature
So if they are much cooler than the surface, they will absorb higher-energy photons emitted from the surface then re-emit them out to space as lower-energy photons
This acts to trap energy in the climate system
Since clouds are almost always colder than the surface, or at most roughly the same temperature, the result is that the LW CRE is ~always positive
Dependence of LW, SW, and net CRE on cloud type:#
Low clouds:
nearly as warm as surface, so little LW trapping: LW CRE \(\approx0\)
So SW reflection wins out, resulting in negative net CRE: low clouds act to cool Earth
High clouds:
much colder than surface, so strong LW trapping: LW CRE\(>0\)
Also tend to be ice and wispy compared to thicker, low clouds, so SW reflection isn’t as strong
So LW wins out over SW, resulting in positive net CRE: high clouds act to warm Earth
Cumulonimbus:
deep towers, so both very strong SW reflection and LW absorption
the SW and LW tend to nearly cancel within the cumulonimbus tower, leading to net CRE\(\approx0\) (albeit slightly positive)
(outside the core tower: cumulonimbus anvils are large contributor to total high cloud amount)
Total, combining all clouds: SW CRE ~ -50 W/m2, LW CRE ~+30 W/m2, so net CRE ~ -20 W/m2
Moist convection#
moist static energy: \(h=c_pT+gz+L_vq\), a.k.a. MSE
adiabatic: MSE conserved
buoyant near surface air + approximately moist adiabatic ascent (i.e. moist static energy unchanged), so a cloud will form near the surface and extend vertically to the level above where the MSE once again equals the near-surface MSE
Cloud microphysics#
Cloud radiative properties depend both on total amount of liquid and ice water but also how that water is distributed into individual cloud droplets and cloud ice particles
Shortwave: cloud albedo increases as cloud droplet radius decreases due to (1) greater total droplet surface area and (2) closer in size to SW wavelengths, which makes scattering more effective
Longwave: less dependence on droplet radius. Instead, total amount of liquid or ice is what matters.
cloud condensation nuclei: an aerosol needs to be present for water vapor to condense onto, forming a cloud droplet.
More aerosols present = more, smaller cloud droplets (for a fixed total amount of water)
Cloud feedbacks#
Net cloud feedback reflects the net effect of all changes to clouds per Kelvin warming
Lots of processes on lots of scales, many of them unresolved by climate models, makes them very uncertain
Best estimate is slightly positive (amplifying) global-mean feedback
But large uncertainty and in fact single largest contributor to uncertainty in net climate feedback parameter and thus in equilibrium climate sensitivity
Uncertainty sources, c.f. Hawkins and Sutton 2007 (2025/3/12)#
climate model: numerical models that simulate Earth’s climate, run on supercomputers, which we use to project future warming
aerosol: microscopic solid particles floating in the atmosphere. E.g. dust, sea salt (both natural) and sulfates from combustion (anthropogenic). Varies by species, but on net they reflect SW back to space, thus acting to cool the climate.
internal variability: natural wiggles, from day-to-day weather to decade-to-decade warmer or cooler periods. These are not caused by human activities; they are intrinsic features of the climate system. Hence they are also called natural variability.
model uncertainty: for a given forcing, how much spread there is across simulations by different climate models in the response
scenario uncertainty: what will society do in the future? Will society decarbonize rapidly? If yes, all else equal, we’d expect less warming compared to if, instead, society further accelerates its use of fossil fuels.
Sea level rise (2025/3/19)#
sea level rise (SLR)
land ice: glaciers and ice sheets
sea ice (doesn’t change sea level)
steric sea level rise: SLR due purely to thermal expansion, not change in total water volume
Global vs. local sea level rise#
Global mean sea level rise causes:
thermal expansion: steric sea level rise
melt of ice sheets and mountain glaciers: the melted water runs off eventually into the ocean
changes in land water storage: dams increase it, groundwater pumping decreases it
Local sea level change causes:
Atmospheric pressure loading
Wind stress
Storm and tidal surges
Coastal erosion and land subsidence
Post-glacial rebound and isostatic adjustment
Gravitational effects of ice sheet melt
Sea ice (2025/4/2, 4/9)#
Note
The sea ice material that was covered in the HW assignment of group/individual presentations will not appear on the exam.
Basics#
sea ice: ice floating in sea or ocean. As opposed to land ice.
Recall: land ice melt contributes to sea level rise, sea ice melt does not
Consists of many individual ice floes, ranging from a few cm to 100s of km
Very well observed in the satellite era = since 1979
Metrics#
sea ice area: area covered by sea ice
sea ice age: how long ago the sea ice formed
sea ice concentration: fraction of a given gridbox that is covered by sea ice
sea ice extent: area spanned by gridboxes with sea ice concentratrion >=15%
sea ice thickness: vertical span of the sea ice from its bottom to its top
Observed trends, Arctic vs. Antarctic#
Arctic: all metrics show clear trends toward decreasing ice: less area, thinner, younger, etc.
Antarctic: no clear trends over satellite era as a whole
Future projections, Arctic vs. Antarctic#
Both similar to observed trends: strong evidence for continued future decreases for Arctic, vs. a much more mixed signal for Antarctic
Using climate model control runs to assess role of natural variability in observed trends#
Motivating question: how likely or not is it that the observed ~40-year Arctic sea ice trends were actually due to internal variability—rather than forced by anthropogenic increases in greenhouse gas concentrations?
In a control simulation, all greenhouse gas concentrations and other boundary conditions are fixed, unchanging. And the simulation is run for hundreds or thousands of years, so that you get a lot of data.
As such, any variations that arise in the simulation must be due to internal variability.
So one can examine the distribution of ~40-year trends in Arctic sea ice in the long-running control simulation. Most will be near zero, and larger trends, both positive and negative, occur much less frequently.
If the observed trend falls within the model’s range of naturally caused trends, that suggests that the observed signal could be mostly due to natural causes.
Whereas if the observed trend falls far outside the model’s distribution, that suggests that the observed signal is most likely due to the anthropogenic forcing.
For the observed Arctic sea ice trend, this exercise indicates that the trends are indeed far beyond what could be credibly explained solely by natural variability.
Tropical cyclones (2025/4/23)#
tropical cyclone, a.k.a. TC.
Regional names: a.k.a. hurricane for North Atlantic and NE Pacific, a.k.a. typhoon in NW Pacific
Spin counterclockwise in northern hemisphere and clockwise in southern hemisphere
Formation#
Coriolis parameter (\(f=2\Omega\sin\varphi\), where \(\varphi\) is latitude): tells you the amount of rotation the spinning Earth is imparting in the horizontal direction at the given latitude. Why does it vary with latitude this way?
at the pole: surface is exactly perpendicular to the rotation axis, so Earth’s spin is exactly aligned with the horizontal plane
at the equator: surface is exactly parallel with the rotation axis, so Earth’s spin does not project onto the horizontal plane at all.
in between: at an angle, so it’s in between
TCs form due to warm, moist air over the tropical oceans that gets organized into big spinning systems via the Coriolis effect
This is why TCs never form in the band directly around the equator: Coriolis is too small there
Note
2025 spring: I’ve decided not to include the Carnot cycle material on the final exam. I’m leaving it here in case you just want to review it for your own sake, but there will be no questions on it.
We can usefully approximate them as Carnot cycle heat engines, with each of the 4 stages:
isothermal expansion: inflow near surface at fixed temperature but gains heat from water vapor evaporated from ocean
adiabatic expansion: ascent concentrated at the eyewall, roughly adiabatic, expanding and cooling (and raining) as it goes up
isothermal compression: outflow near the tropopause loses energy through emitting LW
adiabatic compression: descent in the outer region of the TC, compressing and heating up as it sinks to higher pressures
Hurricane trends and future projections#
In short, trends are fairly weak and future uncertainty is very large!
(This is one of most acute instances of discrepancies between the mainstream scientific consensus regarding climate change and how the public messaging regarding climate change.)
Atmospheric circulation (2025/4/30, 5/7)#
Coordinates and velocities#
Horizontal coordinates and names:
Longitude: east-west = zonal
Latitude: north-south = meridional
Velocity components:
\(u\) for zonal wind, positive eastward
\(v\) for *meridional wind$, positive northward
Velocity direction name conventions:
southerly means from the south: \(v>0\), northerly means from the north: \(v<0\),
westerly means from the west: \(u>0\), easterly means from the east: \(u<0\)
Averages:
zonal average, a.k.a. zonal mean: At each latitude, average the values over all longitudes
time average, a.k.a. time mean: average over the values at all times
Earth spins from west to east
Main circulation features#
Hadley cells: time-mean, zonal-mean overturning circulation in the tropics
Intertropical Convergence Zone (ITCZ): intense rain band in the deep tropics driven by the upward motion of warm, moist air in the ascending branch of the Hadley cells
jet streams: in the upper troposphere subtropics/mid-latitudes, time-mean, zonal-mean fast, westerly flow (up to ~40 m/s)
eddies: everything other than the time-mean, zonal-mean circulation. For our purposes, the ones that matter most are the big traveling storms in the extratropics called “baroclinic eddies”
Angular momentum#
Basics#
angular momentum per unit mass = (distance from rotation axis) x (linear speed in m/s in the direction of rotation)
angular momentum conservation: unless there is a torque, angular momentum is conserved. So moving closer to rotation axis requires spinning up, and moving away from rotation axis requires spinning down
For air in Earth’s atmosphere, two contributions to its angular momentum:
angular momentum of the solid Earth at the given latitude. Equator is farthest from the rotation axis, and the poles are directly on the rotation axis.
angular momentum due to the air’s motion relative to the solid Earth. This depends solely on its zonal velocity \(u\). Since Earth spins west to east, \(u>0\) corresponds to positive angular momentum (meaning same sign as Earth’s angular momentum).
thin shell: vertical extent of Earth’s atmosphere is much smaller than radius of solid Earth. So with respect to angular momentum conservation etc., we ignore vertial motion within the atmosphere, and worry only about latitude.
Using angular momentum and Hadley cells to understand the jet streams#
Air ascends near equator and moves poleward in upper troposphere of the Hadley cell upper branches
Away from surface, very little friction (and thus very little torque). So angular momentum roughly conserved.
Air moves closer to the rotation axis as it moves poleward. To conserve angular momentum, this requires its flow to become more westerly: \(u\) must increase.
This continues all the way to the subtropics where the Hadley cells terminate in either hemisphere, resulting in each hemisphere’s jet stream.
Eddy transports of angular momentum#
Important: you will not be tested on the “Reynolds decomposition” formal equation decomposing into mean meridional circulation and eddy terms.
But you do need to understand the following conceptually:
The angular momentum transports just described were by the Hadley cells, which are the time-mean, zonal-mean circulation
Eddies can also transport angular momentum and other tracers like moisture and heat
For example, in the rotating tank with the cold source in the middle, the eddies moved warm water toward the middle and moved cold water toward the outside. Both of those act to transport heat inward, toward the cold source.
Mean precipitation change (2025/5/7)#
Time-mean atmospheric water vapor budget#
There must be a balance between the source, sink, and transport terms. Specifically, at any given location, these must sum to zero:
precipitation. This is the sink term: atmosphere loses water molecules when they get rained/snowed out
evaporation. This is the source term: atmosphere gains water molecules when they get evaporated out of the surface into the atmosphere
water vapor divergence: This is the transport term: if winds converge water vapor into the region, it acts to increase the amount of water vapor. If conversely winds diverge water vapor out of the region, it acts to decrease the amount of water vapor.
We can create a combined net source term by subtracting evaporation from precipitation: \(P-E\).
If \(P-E>0\), more water vapor is lost through precip than is gained through evaporation. The circulation must therefore converge water vapor into the region to maintain balance.
If \(P-E<0\), less water vapor is lost through precip than is gained through evaporation. The circulation must therefore diverge water vapor out of the region to maintain balance.
In terms of surface “wetness”, we think of \(P-E>0\) as wet regions with lots of rainfall, such as the deep tropics and mid latitudes, and we think of \(P-E<0\) as dry, desert regions as mostly occur in the subtropics.
Simple but powerful scaling argument for future \(P-E\) change.#
This is known as rich-get-richer: a.k.a. wet-get-wetter, dry-get-drier; a.k.a. thermodynamic scaling.
Big picture: wet areas—meaning areas where \(P-E>0\) in the original, unperturbed climate—will tend to get even wetter, i.e. \(\delta(P-E)>0\), as the planet warms. Conversely, dry areas—meaning areas where \(P-E<0\) in the original, unperturbed climate—will tend to get even drier, i.e. \(\delta(P-E)<0\), as the planet warms.
This simple scaling captures the changes in zonal-mean \(P-E\) projected by climate model simulations under increased CO2 surprisingly well: broadly, \(\delta(P-E)>0\) in the deep tropics and mid-latitudes vs. \(\delta(P-E)<0\) in the subtropics.
Chain of argument:
Relative humidity doesn’t change much, so change in specific humidity dictated by Clausius-Clapeyron: increases by ~7% / K warming
Surface temperature change, \(\delta T\), is fairly uniform
Changes in moisture transport divergence/convergence—which, recall, must balance any change in \(P-E\)—is dominated by changes in near-surface specific humidity. (In other words, we neglect the influence of changes in specific humidity higher up in the atmosphere, and we also neglect the influence of changes in the wind fields.)
Combined, these lead to \(\delta(P-E)=\alpha_{CC}(P-E)\delta T\), where
\(\delta(P-E)\) is the change in \(P-E\)
\(\alpha_{CC}\approx 7\%\)/K comes from Clausius-Clapeyron
\(\delta T\) is the global-mean surface air temperature change
\((P-E)\) on the right-hand-side is the original value of \(P-E\) prior to warming