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 δQ v.

Motivation With 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 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: Y t = X t + n t, 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

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, κ xy = 0 If Y is a linear function of X (Y t = aX t ); or a lag shift of X, then κ xy = 1 If Y is a linear function of X and a random white noise, e.g. Y t = aX t + n t Then

Data ERA-Interim Reanalysis (2000 –2012) T, QV and U at 600hPa averaged within two regions  R1 – 15S-5S;5E-15E  R S-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 Coherent periods are between ~10-20 days Spectra are out-of- phase U / 600hPa

Results: Region 1 U / Q 600hPa Shows that easterly winds are associated with drier Coherent periods are also between ~10-20 days Similar results for Region 2

Results: Region 1 U / 600hPa  Averaged between U / Q 600hPa 

Results: Reg. 1 & 2 U—R1 / 600hPa  U—R1 / Q V 600hPa 

Summary The problem is to use cross-spectra analysis to understand the relationship between mid-level U, and T/Q v ; 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 days. The coherence is higher between U and Q V than with T, within the same region. However, coherence is higher between U and downstream T (Region 2), than with Q V.

Cross-spectra analysis of mid-tropospheric thermodynamical variables during southern Africa biomass season. MPO 524

Algorithm flowchart Prepare the two time series (X and Y) Remove the sample means and trends. “Tapering” the first and last 10% of the time series by multiplying with a cosine curve. Perform the FFT (X^ and Y^) Calculate the co-, quadrature coherency and the phase spectra.

Results: Reg Average

Results: Reg

Results: Reg. 1 & 2