Assignment

=Homework 1: Some basic statistics and data handling on the sphere=

Everyone will do the basics.
====Students who are not one of the 5 HW Wiki hosts, or ~5 team leaders, can earn extra credit by doing advanced parts, and/or translating basics into a new language (or making a page to show the Web-tools way to do it). All extra work can count toward your grade, which you will propose to me at the end of term. You may want to keep a list of links or notes about your efforts on a personal side page in the class exam and grading wiki, or privately.====

=Assignment:= =Basic parts:=

1. Download and read, or use OpenDAP to read into memory, the 3-4 variables in 3 dimensions from "Getting Data" page at left. These are all monthly climatologies (~25 year long term means, 'ltm' in file names) of near-surface variables. Array dimensions are [72 or 73, 144, 12], with 2.5 degree and monthly resolutions.
 * Near-surface wind (u and v)
 * Near-surface air temperature (called 'air')
 * Precipitation rate P
 * Outgoing Longwave Radiation (OLR)

2. Calculate zonally averaged P and v over longitude, and plot the resulting latitude-time series. Put latitude on the y axis, and month on the x axis. Make comparison plots of latitude-time sections at our longitude (~90W). Do we live at a "typical" longitude or is North America unusual? (plots aligned sensiby on the page would be nice)

3. Average air temperature over both lat and lon, to make a 12-month time series. Which season has the warmest global mean surface temperature? Can you understand why?

4. Make a map of the temporal (i.e. seasonal) standard deviation of precipitation, expressed as a percentage of the annual mean precipitation. This might be one definition of a "monsoonal" climate.

5. What is the space-time standard deviation of 'air' (temperature)? Best to build this step by step: find the global annual mean of air^2, the global annual mean of air, compute var = meansquare - mean^2, then stdev = sqrt(var).

=Extra credit parts:=

Note: When you decide to tackle an extra credit part, put your name by it, so that

 * ====a. Emily (HW1 host) can find it to show the class the results at the due date====
 * ====b. Other students won't choose the same thing====
 * ==== Pick something unique: maximum of 1 student per extra credit question in a given language ====

Who? language?
a) How does the south Florida gridpoint stack up globally by the monsoon measure of Basic Question 4 above? To answer this, sort the gridpoints in the array plotted in Q4. What percentile (quantile) are we in? Top 5%? Top 10%? etc.

b) Make a map of the temporal skewness of P, P'^3 t where P' means the deviation from a time average: P'(x,y,t) = P(x,y,t) - P t(x,y). Where is it large? Perhaps monsoonal deserts, where there are a lot of months with P=0 values, then some strong rains in one season? Pick a region where it is large and make a histogram of P. If you pick one gridpoint, your histogram will just have 12 values, so it may make sense to pool several nearby gridpoints and get a fuller histogram.

b) Ramage's classic (1960s textbook) definition of a "monsoon" climate is that the direction of the mean wind varies by more than 120 degrees from winter to summer. Compute wind direction in January and July and make a map of where this criterion is satisfied. (Careful! Winds from 350 and 10 degrees are only 20 degrees apart. You may need the modulo operation.)

Who? language?
a) Make a synthetic Gaussian patch SST anomaly SSTA(lat,lon) = A *exp(- (lat^2/DY) - (lon-lon0)^2/DX ). This is a Gaussian patch centered on the equator at lon0, with lat & lon scales DY and DX. Pick some modest scale (maybe 10 degrees lat by 30 lon) (You will first want to build 2D arrays of lat and lon to put into the above formula.)

b) Look up the Gill solution for the linear wind anomalies driven by an atmospheric heat source with this structure. Build anomalous wind fields uA(lat,lon) and vA(lat,lon). These needn't be exact. Make a contour plot of SSTA with vectors of {uA, vA} on it. This might be a good GrADS project.

c) Now add this {uA, vA} (lon0) to the mean wind pattern {u,v}, shifted to different longitudes and in different seasons. Start with one longitude and season, then later you can loop over all months and various longitudes. Compute the WIND SPEED ANOMALY field sqrt((u+uA)^2 + (v+va^2)).

d) Suppose the main feedback effect is via wind speed (WS) effects on evaporation: increased wind speed will damp an SST anomaly, decreased wind speed will amplify it. Make a scalar measure of this feedback metric F: the SSTA-weighted mean WS anomaly (you can just do a global mean, since the SSTA weighting and WS anomaly will all be tiny in regions far away from the patch).

e) Make a contour plot of F(lon0, month). At what longitudes and seasons are tropical SST anomalies most damped by this effect? Least damped (or amplified)?

Who? language?
It has been calculated that an instant CO2 doubling would reduce global longwave emission OLR xyt by 4 Wm-2. Restoration of climate equilibrium might occur through changes to the emitting temperature Te of the Earth, to restore global mean OLR xyt to its present value (we assume this global OLR xyt balances absorbed solar radiation ASR xyt, and let's assume that doesn't change).

As a thought exercise, estimate some ways the emission temperature of the Earth could change to increase OLR xyt by 4 Wm-2: first through mean warming, then through changes in the space-time distribution of Te, via the 4th power nonlinearity in the Stefan-Boltzmann equation OLR = SB * Te^4 xyt, where SB = 5.67e-8 in MKS units.

a) What is the emitting temperature Te(lat,lon,mo) implied by the OLR(x,y,mo) dataset? What is its global mean? Te xyt

b) How much would Te xyt have to be increased, if its increase were uniform over the globe and across seasons, to increase OLR by 4 Wm-2?

c) What is the contribution of the observed space-time fluctuations Te' = Te(lon,lat,mo) - Te xyt on global mean OLR xyt? To answer that, make a plot of OLR xyt(A), where A is the amplitude by which one rescales Te fluctuations: Te_test = ATe' + Te xyt . Since Te' xyt=0, rescaling the fluctuations doesn't change the global mean. What value of A causes OLR xyt to increase by 4 Wm-2? What are the coldest or warmest value of Te_test for that value of A? (Are these physically absurd?)

Who? language?
Consider one of the datasets given here - perhaps a wind component. Represent it as a floating point (4Byte) array.

a) Compute its variance in some dimension(s) as var = meansquare - mean^2. Take the square rood (stdev), to make the units physical.

b) Now rescale the physical variable, as if you were changing its units (say, converting wind to mm/century or km/s). Compute the variance again, expressed in MKS units. How inaccurate is the calculation, as a function of your units rescaling parameter?

c) Express your results in a plot of calculated standard deviation vs. your rescaling parameter, with lines for both 8B and 16B (double precision) variables. What is the safe range for the size of numbers in such computations?

d) Do the same for skewness (3rd moment) or kurtosis (4th).