# Meaning of $\frac{1}{\sqrt{dt}}$ in stochastic forcing

I am running a 2D fluids simulation with a stochastic forcing $$f$$ in a doubly-periodic box, i.e. solving

$$\frac{\partial \nabla^2 \psi}{\partial t} = J(\psi,\nabla^2 \psi) +f,$$ where $$J$$ is a Poisson bracket.

The forcing I've chosen is of the form

$$f= \sum_{k,l} c \sin(k x + \alpha_k) \sin(l y + \beta_l),$$ where the wavenumbers $$k$$ and $$l$$ are selected from a thin annulus centered at $$k_f$$ (a common prescription), and $$\alpha_k$$ and $$\beta_l$$ are phases which are in some sense stochastic (more on this later). (When $$k=0$$ or $$l=0$$ one needs a slightly different treatment but this is a minor detail).

One can show that $$\int d^2 x \, \psi \frac{\partial \nabla^2 \psi}{\partial t} = - \frac{\partial E}{\partial t}$$ where $$E$$ is the total kinetic energy. We can use this to compute the energy injection rate from the forcing:

$$\varepsilon = -\langle f \xi \rangle = \frac{N c^2}{4 k_f^2},$$ where $$\nabla^2 \xi = f$$ and $$N$$ is the number of wavevectors in the annulus. This motivates the choice $$c = 2 k_f\sqrt{\frac{\varepsilon}{N}}$$.

Now, we need to prescribe a temporal correlation for the forcing. The standard choice is white noise, i.e. $$f$$ is delta-correlated in time. In (for example) Appendix A of Srinivasan and Young (2012), the authors select the phases i.i.d. from a uniform distribution and aver that the forcing must be normalized by $$1/\sqrt{\delta t}$$ ($$\delta t$$ being the time-step of the integration algorithm) in order to ensure that it is delta-correlated. This raises two questions with which I'm struggling:

1. How, precisely, does this lead to delta-correlated forcing? I'm having some trouble showing it analytically.
2. What now of the energy injection rate? Isn't it altered by a factor of 1/$$\delta t$$? And aren't the dimensions now compromised?

Furthermore, as is pointed out in the same appendix, in a Runge-Kutta algorithm, the forcing must be kept reasonably smooth during the course of a time-step, so in that paper, they select the phases within a time-step by linear interpolation. I'm finding this tricky to implement with the library I'm using, so I had the idea to instead update the phases by their own random walk: $$\alpha_k(t+\delta t) = \alpha_k(t) + \sqrt{\delta t} \eta$$ with $$\eta\sim {\cal N}(0,\sigma^2),$$ and the same for the $$\beta_l$$. Then one can show that this leads to the correlation function

$$\langle f(\mathbf{x},0)f(\mathbf{x},t) \rangle = \frac{Nc^2}{4} \exp(-|t|/\tau)$$ where $$\tau= 1/\sigma^2$$. This forcing is nice in that it's nice and smooth and you can control the correlation time...unless you want white noise. Thus, a third question:

1. Can this forcing prescription be adjusted simply so that, in the limit $$\tau \to 0$$, the forcing is temporally white? Does, say, normalizing $$f$$ by $$1/\sqrt{\tau}$$ work?

Thanks in advance to anyone who can help me with this.

• To answer (what I understood is) the first point, you must make units right. Since $<\eta(t)\eta(t')>\propto\delta(t-t')$ units of $\eta$ are $1/\sqrt{t}$. When you inject it to the stochastic equation $dx=adt + bdt\eta$ the last term is actually proportional to $\sqrt{dt}$ Jun 4 '20 at 1:02
• I've thought about this more and it's easier for me to conceptualize in terms of the central limit theorem, which shows that, if $\partial_t \zeta = \xi(t)$ for some random $\xi$, $\zeta(\tau)-\zeta(0) \sim \sqrt{\tau \delta t} {\cal N} (0, \langle \xi^2 \rangle)$, so we need the normalization to ensure that $\langle \zeta^2 \rangle \sim t$ like a good Brownian motion. I'm still trying to figure out how this affects the rate of energy injection... Jun 4 '20 at 1:38
• I think I now roughly understand the energy injection issue. The rate of energy injection is not, as I wrote, $-\langle \xi f \rangle,$ but $-\langle \psi f \rangle$. It stands to reason, at least in a hand-waving sort of way, that $\psi \sim \sqrt{t} \xi$ (where again $\nabla^2 \xi = f$), so the rate of energy injection in fact is only constant in time once the normalization is introduced. Jun 4 '20 at 1:59
• Reading this paper aip.scitation.org/doi/pdf/10.1063/1.870050 helped me resolve these conceptual problems and identify my error Jun 4 '20 at 2:55
• consider posting it as an answer, with short description of the solution Jun 4 '20 at 5:13

The need for the normalization can be illustrated as follows. Consider only the contribution of the forcing: $$\partial_t \zeta = f(t).$$

If we discretize time in steps $$\delta t$$, $$f$$ is a sequence of iid random variables $$\{f_i\}$$ with mean 0. Then we have (setting $$\zeta(0)=0$$) $$\zeta(t) = \delta t \sum_i f_i.$$

According to the central limit theorem, for $$t\gg \delta t$$, we have $$\sum_i f_i \sim \sqrt{N}{\cal N}(0,\langle f^2 \rangle)$$ where $$N=t/\delta t$$. Thus

$$\zeta(t) \sim \sqrt{t \delta t} {\cal N}(0,\langle f^2 \rangle)$$ and it is evident that $$f$$ must be normalized by $$1/\sqrt{\delta t}$$ in order for its evolution to be independent of the timestep.

My issue with the energy injection rate largely stemmed from an error in calculation. The rate is $$\varepsilon = - \langle \psi f\rangle,$$ not $$-\langle \xi f \rangle$$ (where again, $$\nabla^2 \xi = f$$). Following K. Alvelius PoF 11, 1880 (1999), we have (considering only the forcing's contribution to the dynamics) $$\psi = \int_0^t d\tau \, \xi(\tau),$$ so $$\varepsilon = - \int_0^t d\tau \, \langle \xi(\tau)f(t) \rangle.$$

My previous, erroneous calculation did not have the correct units, and clearly missed a factor with dimensions of time. In the case of my implementation of the forcing, the energy injection rate follows as

$$\varepsilon =\frac{Nc^2}{4k_f^2} \tau_c (1- \exp(-|t|/\tau_c)$$.

After many correlation times, the second term is negligible and choosing $$c= 2 k_f \sqrt{\frac{\varepsilon}{N \tau_c}}$$ fixes a chosen rate $$\varepsilon$$. Note that factor $$1/\tau_c$$, previously missed. This normalization also ensures that in the limit $$\tau_c \to 0$$, the volume of the correlation function is correctly conserved and a delta function is obtained, as desired.