I applied a discrete wavelet transform to horizontal wind speed data to receive the below plot. I'm basically trying to use the information from the detail coefficient (the turbulent flow) for further analysis, but I'm not sure the best direction to go in. I don't have much experience with Wavelet Transform, so forgive me if there are obvious options, but the examples I've seen usually discard the higher frequency information since it's the noise of the signal. Is there anything further I can do with this discrete wavelet transform like statistic analysis or forecasting?
The path to pursue really depends on the question that you are trying to answer.
First of all, I would suggest double checking that your DWT is actually doing what you expect it to do. The plot that you shared suggests that it is successful in separating the low frequency coherent (laminar?) flow from the high frequency turbulent flow, but it would be helpful to figure out which frequencies are present in the high frequency component in order to confirm that the processing parameters (e.g. decomposition level) were properly chosen.
Once convinced that your wavelet decomposition provides you with useful information about the turbulent flow, what should you do with these high pass filtered data? I suggest computing their variance over 1 hour long intervals. This is a measure of the "energy" of the signal over the chosen interval. If you are dealing with large amounts of data this would allow you to boil down your time series into a single sample per hour. Maybe you will be able to spot diurnal variations in the turbulent flow (e.g. maybe turbulent flow is higher at dawn). If you have multiple stations it would be interesting to study if the turbulence variations share the same behavior.
Before venturing into time series forecasting, I would really take a closer look at you data and try to identify trends or nail down possible outliers.
Last but not least, I would suggest posting your question on Physics Stack Exchange (e.g. https://physics.stackexchange.com/) rather than on SO.