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Cross-spectra analysis of mid-tropospheric thermodynamical variables during Southern Africa biomass season.
Yemi Adebiyi
MPO 524
Motivation In southeast Atlantic (at
~600 hPa), there is a significant correlation of increased zonal winds, with cooling and moistening anomalies during polluted condition (tau>0.2)
… For a biomass season between July-October.
Maximum correlation between δU and δT occurs a day before maximum correlation δU and δQv.
MotivationWith the entire mid-level system moving at about 5-7deg/day westwards, this correlation implies a downstream cooling.
• What is the dynamical relationship?
• Are the associated time series coherent? – Cross-spectra analysis
Previous study: MJO Madden and Julian 1971,
used the cross-spectra analysis to support the detection of oscillation in the zonal winds of the Tropical pacific.
An easterly wind at 850hPa will be accompanied by low surface pressure and a westerly wind at 150hPa at a period between 30-90 days.
This turns out to be Madden-Julian Oscillation.© Madden and Julian, 1971
Suppose we have two time series X(t) and Y(t), t=1,……N,
Then the cross-covariance function:
If X and Y are linearly related: Yt = Xt + nt, then the cross-correlation would be:
Now in the frequency domain, we can take Fourier transform of the cross-covariance, to give the cross-spectrum:
Cross-Spectra Analysis
Since the cross-spectrum is generally a complex function, it can be represented in two ways:
1. It can be decomposed into real and imaginary parts
2. It can be written in polar coordinate:
Cross-Spectra Analysis
Cross-Spectra Analysis
The (squared) coherency spectrum can be defined as:
This is similar to the (squared) correlation coefficient.
Properties:
• For a completely random variable X and Y, κxy = 0
• If Y is a linear function of X (Yt = aXt); or a lag shift of X, then κxy = 1
• If Y is a linear function of X and a random white noise, e.g. Yt = aXt + nt
Then
Data ERA-Interim Reanalysis
(2000 –2012)
T, QV and U at 600hPa averaged within two regions R1 – 15S-5S;5E-15E R2 -- 15S-5S;10W-0E
July and October (Biomass season) Remove the sample means
and trends. Employ tapering to reduce
leakages.
Time Series
Region 1 Region 2
Results: Region 1
Shows that easterly winds are associated with cooler air @600hPa
Coherent periods are between ~10-20 days
Spectra are out-of-phase
U / T @ 600hPa
Results: Region 1U / QV @ 600hPa
Shows that easterly winds are associated with drier air @600hPa
Coherent periods are also between ~10-20 days
Similar results for Region 2
Results: Region 1
U / T @ 600hPa
Averaged between 2000-2012
U / QV @ 600hPa
Results: Reg. 1 & 2
U—R1 / T—R2 @ 600hPa
U—R1 / QV—R2 @ 600hPa
Summary The problem is to use cross-spectra analysis to
understand the relationship between mid-level U, and T/Qv; within the same region (and downstream).
The result shows that, within the same region, easterly winds are associated with cooler and moister air, with periods between 10-20days.
The coherence is higher between U and QV than with T, within the same region.
However, coherence is higher between U and downstream T (Region 2), than with QV.
Cross-spectra analysis of mid-tropospheric thermodynamical variables during southern Africa biomass season.
MPO 524