Saturation and the “universal” spectrum for vertical profiles of horizontal scalar winds in the atmosphere

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Abstract
A theory is presented which explains the universal nature of one‐dimensional vertical wave number, k, power spectral densities (PSDs) of horizontal winds as measured in the atmosphere and predicted by VanZandt. The theory is that the PSD amplitude at any given wave number (greater than a certain minimum, k*) is determined by its saturation value due either to shear instability (i.e., critical Richardson Number) or, more likely, to convective instability. This explains why the PSD amplitudes observed do not grow exponentially with increasing altitude. This saturation theory assumption plus other considerations leads to a PSD of the form N2/kn, where n is in the range of about 2.5 to 3 and N is the Brunt frequency. A simplified model involving superimposed narrow bands of gravity waves as well as a model based merely on dimensional arguments both lead to n = 3. The full model not only explains the observed spectral slopes but also predicts the PSD amplitude in the troposphere to be 3.5 times smaller than in the stratosphere. The derivation of the model is based on the saturation condition that ∫ k2PSD(k) dk = N2. The model may also apply to the ocean and explain the Garrett‐Munk vertical wave number spectrum.