The gas velocity is unknown since no exact analytic solution for turbulence in a disc – and turbulence in general – are known. However, statistical properties of turbulence can be inferred from laboratory, numerical experiment or theory, and turbulent fluctuations can be modelled using stochastic processes, independently from the origin of the turbulence itself. In a seminal study, Thomson (1987) proved that the only expression of vg that is consistent with Kolmogorov turbulence and the hydrodynamical equations is
(16)$$\begin{eqnarray*}
\frac{\mathrm{d}v_{\rm g}}{\mathrm{d}t} = -\frac{v_{\rm g}}{t_{\rm e}} + \frac{\sqrt{D}}{t_{\rm e}}\dot{w} ,
\end{eqnarray*}$$where te denotes the Lagrangian time-scale of the turbulence, D is the turbulent diffusivity (in units m2s−1). w is a Wiener process, such that its derivative is a white noise such that
(17)$$\begin{eqnarray*}
\left\langle \dot{w} (t) \right\rangle = 0,
\end{eqnarray*}$$(18)$$\begin{eqnarray*}
\left\langle \dot{w} (t) \, \dot{w} (t^{\prime }) \right\rangle = \delta (t - t^{\prime }) ,
\end{eqnarray*}$$where δ denotes the Dirac distribution and the notation · > is the expectation operator (see also Sawford 1984; Wilson & Sawford 1996). Equation (16) describes turbulent fluctuations from a Lagrangian point of view (Taylor 1922). From equation (16), the gas velocity can be rewritten
(19)$$\begin{eqnarray*}
v_{\rm g} = \zeta \left(t,t_{\rm e},D \right) ,
\end{eqnarray*}$$where ζ is a stationary Ornstein–Uhlenbeck process defined by
(20)$$\begin{eqnarray*}
\left\langle \zeta (t,t_{\rm e},D) \right\rangle = 0,
\end{eqnarray*}$$(21)$$\begin{eqnarray*}
\left\langle \zeta (t,t_{\rm e},D) \, \zeta (t^{\prime },t_{\rm e},D) \right\rangle = \frac{D}{2 t_{\rm e}} \rm {e}^{-\frac{\vert t - t^{\prime } \vert }{t_{\rm e}}} .
\end{eqnarray*}$$Equation (16) defines a model of turbulence with two parameters, D and te. In discs, te is typically of order one orbital period, since turbulent vortices are stretched out by differential rotation in a few orbits (e.g. Beckwith, Armitage & Simon 2011). From equation (21), D is related to the autocorrelation of the turbulent noise according to
(22)$$\begin{eqnarray*}
D = 2 \int _{0}^{+\infty } \left\langle v_{\rm g}\left(0 \right) v_{\rm g}\left(t \right)\right\rangle \mathrm{d}t .
\end{eqnarray*}$$Equation (22) can alternatively be seen as a definition of the turbulent diffusivity, useful in practice to measure D in numerical simulations. The Wiener–Khinchin theorem ensures that the power spectrum of the turbulent velocity field S(ω) is the Fourier transform of this autocorrelation function, i.e.
(23)$$\begin{eqnarray*}
S \left(\omega \right) = \frac{1}{2 \pi } \int _{-\infty }^{+\infty } \mathrm{e}^{- i \omega t} \left\langle v_{\rm g}\left(0 \right) v_{\rm g}\left(t \right)\right\rangle \mathrm{d}t = \frac{D}{2\pi \left(1 + \omega ^2 t_{\rm e}^2 \right)} .
\end{eqnarray*}$$Thus, in the inertial subrange ($\omega ^2 t_{\rm e}^2 \gg 1$), we have S(ω) ∝ ω−2, whose equivalent in the wavelength space is $\tilde{S}(k)\propto k^{-5/3}$ (Batchelor 1950). From equation (23), the standard deviation of the velocity fluctuation σ is
(24)$$\begin{eqnarray*}
\sigma ^{2} \equiv \int _{-\infty }^{+\infty } S\left(\omega \right) \mathrm{d} \omega = \frac{D}{2t_{\rm e}} .
\end{eqnarray*}$$Physically, equation (24) is a turbulent fluctuation–dissipation theorem.
Citation: Taylor 1922
Citation function: Uses