How to calculate the kinetic energy spectra from ocean current timeseries?

Question

How does one practically compute the kinetic energy spectra from a timeseries of u- and v- Cartesian components of ocean current velocity ($m s^{-1}$) obtained from an Acoustic Doppler Current Profiler (ADCP) moored on the seabed?

Do you:

1) compute the spectra of each u- and v- components and then calculate the resulting kinetic energy from the spectra in the frequency domain? i.e. $KE(f) = 0.5\times(u_{(f)}^2 + v_{(f)}^2)$; where $KE_{(f)}$ is the kinetic energy spectrum per unit frequency and similarly $u_{(f)}$ and $v_{(f)}$ are the u- and v- power spectra?

or;

2) compute the kinetic energy $KE = 0.5\times(u_{(t)}^2 + v_{(t)}^2)$, where $u_{(t)}$ & $v_{(t)}$ are the current velocity timeseries and then calculate the power spectral density (i.e. using Matlab'spwelch) of the resulting KE?

Any guidance with the theory or practical calculation would be very much appreciated.

Thank you.

Answer 1

Do you get much different results if you do both ways?

I think if you want the kinetic energy spectrum, use your second method. Some separately compute velocity and KE spectra, e.g., figure 2 ofhttp://journals.ametsoc.org/doi/pdf/10.1175/2009JPO4330.1(DP Wang et al, JPO, Apr 2010). Comparing velocity and KE spectra, the KE spectra looks similar to the more powerful velocity spectrum, especially at low frequencies.

Others use your first method (e.g.,http://journals.ametsoc.org/doi/pdf/10.1175/JPO-D-15-0087.1, Rocha et al, JPO, Feb 2016)