# Stochastic Forcing¶

There is an option to apply a stochastic force field.

See Nyx/Exec/DrivenTurbulence for an example; note that

nyx.do_forcing = 1


must be set in the inputs file.

The external forcing term in the momentum equation ([eq:momt]) is then given by

${\bf S}_{\rho \Ub} = \rho_b \mathbf{f}$

where the acceleration field $$\mathbf{f}(\mathbf{x},t)$$ is computed as inverse Fourier transform of the forcing spectrum $$\widehat{\mathbf{f}}(\mathbf{k},t$$). The time evolution of each wave mode is given by an Ornstein-Uhlenbeck process (see :raw-latex:\cite{SchmHille06,Schmidt14} for details). Since the real space forcing acts on large scales $$L$$, non-zero modes are confined to a narrow window of small wave numbers with a prescribed shape (the forcing profile). The resulting flow reaches a statistically stationary and isotropic state with a root-mean-square velocity of the order $$V=L/T$$, where the integral time scale $$T$$ (also known as large-eddy turn-over time) is usually set equal to the autocorrelation time of the forcing. It is possible to vary the force field from solenoidal (divergence-free) if the weight parameter $$\zeta=1$$ to dilational (rotation-free) if $$\zeta=0$$.

To maintain a nearly constant root-mean-square Mach number, a simple model for radiative heating and cooling around a given equilibrium temperature $$T_0$$ is applied in the energy equation ([eq:energy]):

$S_{\rho E} = S_{\rho e} + \Ub \cdot {\bf S}_{\rho \Ub} = -\frac{\alpha k_{\rm B}(T-T_0)}{\mu m_{\rm H}(\gamma-1)} + \rho_b\Ub\cdot\mathbf{f}$

The parameters $$T_0$$ and $$\alpha$$ correspond to temp0 and alpha, respectively, in the probin file (along with rho0 for the mean density, which is unity by default). While the gas is adiabatic for $$\alpha=0$$, it becomes nearly isothermal if the cooling time scale given by $$1/\alpha$$ is chosen sufficiently short compared to $$T$$. For performance reasons, a constant composition (corresponding to constant molecular weight $$\mu$$) is assumed.

## List of Parameters¶

Parameter Definition Acceptable Values Default
forcing.seed seed of the random number generator Integer $$>0$$ 27011974
forcing.profile shape of forcing spectrum 1 (plane), 2 (band), 3 (parabolic) 3
forcing.alpha ratio of domain size $$X$$ to integral length $$L=X/\alpha$$ Integer $$>0$$ 2 2 2
forcing.band_width band width of the forcing spectrum relative to alpha Real $$\ge 0$$ and $$\le 1$$ 1.0 1.0 1.0
forcing.intgr_vel characteristic velocity $$V$$ Real $$> 0$$ must be set
forcing.auto_corrl autocorrelation time in units of $$T=L/V$$ Real $$> 0$$ 1.0 1.0 1.0
forcing.soln_weight weight $$\zeta$$ of solenoidal relative to dilatational modes Real $$\ge 0$$ and $$\le 1$$ 1.0

Triples for forcing.alpha, forcing.band_width, forcing.intgr_vel, and forcing.auto_corrl correspond to the three spatial dimensions.