where $v_w(z)$ is a wind velocity, and $\beta$ is the zenith angle. The Greenwood frequency can be associated with turbulence temporal error. The atmospheric turbulence conjugation process can be limited by temporal deficiencies as well as spatial ones\cite{karr1991temporal}. Assuming perfect spatial correction, Greenwood\cite{greenwoodbandwidth} showed that the variance of the corrected wavefront due to temporal limits is given by: $$\sigma^2_{temp} = \int_{0}^{\infty} |1 - H(f,f_c)|^2 P(f) df$$ where the $P(f)$ is the disturbance power spectrum\cite{tysonprinciplesbook}. The higher-order wavefront variance due to temporal constraints is $$\sigma_{temp}^2 = [\frac{f_G}{f_{3~dB}}]^{5/3}$$ where the $f_G$ is the Greenwood frequency.
Greenwood frequency as an estimation of a controller's bandwidth
The required frequency bandwidth of the control system is called the Greenwood frequency\cite{greenwoodbandwidth}. An adaptive optics system with a closed-loop servo response should reject most of the phase fluctuations. Greenwood\cite{greenwoodbandwidth} calculated the characteristic frequency $f_G$ as follows:
where $\beta$ is the zenith angle, $v_w$ is wind velocity. In the case of a constant wind and a single turbulent layer, the Greenwood frequency $f_G$ can be approximated by: $$f_G = 0.426 \frac{v_w}{r_0},$$ where $v_w$ as the velocity of the wind in meters/sec and $r_0$ is the Fried parameter. Greenwood\cite{greenwoodbandwidth} determined the required bandwidth, $f_G$ (the Greenwood frequency), for full correction by assuming a system in which the remaining aberrations were due to finite bandwidth of the control system\cite{saha2010aperture}. Greenwood derived the mean square residual wavefront error as a function of servo-loop bandwidth \textit{for a first order controller}, which is given by:
where $f_c$ is the frequency at which the variance of the residual wavefront error is half the variance of the input wavefront, known as 3 db closed-loop bandwidth of the wavefront compensator, and $f_G$ the required bandwidth\cite{saha2010aperture}. It must be noted that the required \textit{bandwidth for adaptive optics does not depend on height}, but instead is proportional to $v_w /r_0$ , which is in turn proportional to $\lambda^{-6/5}$. If the turbulent layer moves at a speed of 10 m/s, the closed loop bandwidth for $r_0 \approx 11$ cm, in the optical band (550 nm) is around 39 Hz\cite{saha2010aperture}.
For most cases of interest, the Greenwood frequency of the atmosphere is in the range of tens to hundreds of Hertz. Beland and Krause-Polstorff\cite{greenwoodfreqvariation} present measurements that show how the Greenwood frequency can vary between sites. Mt. Haleakala in Maui, Hawaii, has an average Greenwood frequency of 20 Hz. For strong winds and ultraviolet wavelengths, the Greenwood frequency can reach 600 Hz. The system bandwidth on bright guide stars is, in most cases, several times larger than the Greenwood frequency.
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