Turbulence driving

The block named &TURB_PARAMS contains the parameters related to the turbulence driving. Originally implemented by Andrew Mcleod.

General principle[1]

The model used for turbulent driving in Ramses is a generalization of the Ornstein-Uhlenbeck process. The force is computed in Fourier space and then applied to the gas. The evolution of the Fourier modes \(\vec{\hat{f}}\) of the force was obtained via the resolution of the following stochastic differential equation:

\[ \mathrm{d}\vec{\hat{f}}(\vec{k}, t) = - \vec{\hat{f}}(\vec{k}, t)\dfrac{\mathrm{d}t}{T} + F_0(\vec{k})\vec{P_\chi}\left(\vec{k}\right) \mathrm{d}\vec{W}_t. \]

In this equation, \(\mathrm{d}t\) is the timestep for integration and \(T\) is the autocorrelation timescale. The perturbation \(\mathrm{d}\vec{W_t}\) is a small vector randomly chosen following the Wiener process. The power spectrum \(F_0\) is a way to select the relevant mode.

Example

A parabolic power spectrum between \(k=1\) and \(k=3\):

\[\begin{split} F_0(\vec{k}) = \begin{cases} 1 - \left(\dfrac{\vec{k}}{2\pi} - 2\right)^2\text{ if } 1 < \dfrac{\vert k \vert}{2\pi} < 3 \\ 0 \text{ if not.} \end{cases} \end{split}\]

The projection operator \(\vec{P_\chi}\) is a weighted sum of the components of the Helmholtz decomposition of compressive versus solenoidal modes:

\[ \vec{P_\chi}(\vec{k}) = (1 - \chi) \vec{P}^{\perp}(\vec{k}) + \chi \vec{P}^{\parallel}(\vec{k}) \;, \]

with \(\vec{P}^{\perp}\) and \(\vec{P}^{\parallel}\) the projection operators respectively perpendicular and parallel to \(\vec{k}\) and \(\chi\) the compressive driving fraction. This compressive driving fraction applies only to the driving and is different from the compressive ratio measured in the velocity field. The forcing field \(\vec{f}(\vec{x}, t)\) is then computed from the Fourier transform:

\[ \vec{f}(\vec{x}, t) = g(\chi) f_{\mathrm{rms}} \int\vec{\hat{f}}(\vec{k}, t) e^{i\vec{k}\cdot x} d^3\vec{k}\;. \]

The parameter \(f_{\mathrm{rms}}\) is directly linked to the power injected by the turbulent force into the simulation. The \(g(\chi)\) factor is an empirical correction so that the resulting time-averaged root-mean-square of the power of the Fourier modes is equal to \(f_{\mathrm{rms}}\), independently of the compressive fraction \(\chi\).

Overview of parameters

Variable name	Fortran type	Default value	Notation	Description
turb	boolean	.false.		Turn on or off driving
turb_seed	integer	-1		Random number generator seed. -1 = random
turb_type	integer	1		How the driving changes over time. 1=driven evolving, 3=decaying
instant_turb	boolean	.true.		Generate initial turbulence before start
comp_frac	float	0.3333	\(\chi\)	The weight of compressive over solenoidal modes
turb_T	float	1	\(T\)	Turbulent velocity auto-correlation time in code units.
turb_Ndt	integer	100	\(T/dt\)	Number of timesteps per auto-correlation time
turb_rms	float	1	\(f_\mathrm{rms}\)	Root-mean-square turbulent forcing in code units.
turb_min_rho	float	1d-50		Minimum density for turbulence. Not forcing is added onto cellswith a density less than this value.
forcing_power_spectrum	string	parabolic	\(F_0\)	Power spectrum type of the forcing, which describes the relative strength of individual modes. Options are: power_law, parabolic, konstandin

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[1]

adapted from Brucy et al. 2024.