Large-scale effects on the decay of rotating helical and non-helical turbulence

Abstract

Decaying three-dimensional (3D) turbulence is studied via direct numerical simulations (DNS) for an isotropic non-rotating flow and for rotating flows with and without helicity. We analyze the cases of moderate Rossby number and large Reynolds number focusing on the behavior of the energy spectrum at large scales and studying its effect on the time evolution of the energy and integral scales for E(k) ∼ k 4 initial conditions. In the non-rotating case we observe the classical energy decay rate t −10/7 and a growth of the integral length proportional to t 2/7 in agreement with the prediction obtained assuming conservation of the Loitsyanski integral. In the presence of rotation we observe a decoupling in the decay of the modes perpendicular to the rotation axis from the remaining 3D modes. These slow modes show a behavior similar to that found in two-dimensional (2D) turbulence whereas the 3D modes decay as in the isotropic case. We phenomenologically explain the decay considering integral conserved quantities that depend on the large scale anisotropic spectrum. The decoupling of modes is also observed for a flow with a net amount of helicity. In this case, the 3D modes decay as an isotropic fluid with a constant, constrained integral length, and the 2D modes decay as a constrained rotating fluid with maximum helicity.

Decay of Batchelor and Saffman rotating turbulence

“…The purpose of simulations A and C is to recover well known results for isotropic ∼ k 2 and ∼ k 4 turbulence, and to use them as a starting point to analyze the more complex rotating cases. Note the decay of ∼ k 4 turbulence (corresponding to runs C and D) was studied in detail in [26], and is considered here to compare with the ∼ k 2 case.…”

“…Numerical simulations and models also show a slow down of the energy decay for many different initial conditions [11,[20][21][22][23][24], together with the trend towards two-dimensionalization [22,25], and the higher cyclonic-over-anticyclonic activity [26,27]. The first theoretical study of the decay of rotating turbulence was reported in [28], and considered the decrease of the energy transfer in the presence of rotation to explain the observed slow down in the decay, and to predict decay rates for the energy.…”

“…For flows with initial integral scale sufficiently smaller than the domain size, the case of initial conditions with energy spectra ∼ k 4 was studied in detail in Refs. [26,27], where different decay rates were reported for the two-dimensional and threedimensional modes. While E 3D was found to decay as in the case of isotropic turbulence, a slower decay was found for E 2D in the intermediate-Rossby range as defined in [25].…”

The decay of turbulence in rotating flows

“…It is worth pointing out that these quantities were also shown to be conserved in other systems: proofs of the conservation of K and I 2D for quasigeostrophic flows can be found in [50]. In practice, these quantities are only approximately conserved in numerical simulations, see e.g., the approximate constancy of I 2D and K reported for rotating flows in [47].…”

Rotating Taylor–Green flow

“…The increase of computational power has also allowed the study of rotating flows by simulations. Most numerical investigations have focused on decaying turbulence (Bardina, Ferziger & Rogallo 1985;Mansour, Gambon & Speziale 1992;Bartello, Metais & Lesieur 1994;Hossain 1994;Squires et al 1994;Godeferd & Lollini 1999;Smith & Waleffe 1999;Morinishi, Nakabayashi & Ren 2001;Müller & Thiele 2007;Thiele & Müller 2009;Teitelbaum & Mininni 2010;Yoshimatsu, Midorikawa & Kaneda 2011), with more recent investigations of forced rotating turbulence both at large scales (Yeung & Zhou 1998;Mininni, Alexakis & Pouquet 2009;) and at small scales in order to observe the development of an inverse cascade (Smith & Waleffe 1999;Teitelbaum & Mininni 2009). Computational cost however did not allow for an exhaustive coverage of the parameter space.…”