7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Non-equilibrium Turbulence Effect on Plumes
Nobumitsu Yokoi1
Institute of Industrial Science, University of Tokyoe-mail (speaker): nobyokoi@iis.u-tokyo.ac.jp

Plumes are persistent vertical fluctuating motions
originated from localized buoyancy source, andubiquitously observed convective elements at highRayleigh numbers. The plume flow direction is along thelarge-scale gradient of density and temperature, in thissense, plumes are similar to streamers in fusion plasmas.
In the stellar convection, negatively buoyant plumes
generated by the surface cooling play a crucial role inturbulent mixing of mass and heat. The properties ofturbulence and its transport are fairly different between
the convection locally driven by the entropy gradient in
the whole region (locally-driven convection) and thecounterpart non-locally driven by the strong cooling at
the surface (non-locally-driven convection).[1,2] Thehorizontal structureof the convective
cells is much larger
in the locallydriven case, andthe plume motions
are prominent at
the near-surface Figure 1: Entropy contours obtained fromregion in the non- DNSs for (a) the locally driven and (b) thenon-locally driven convections. Horizontal
locally driven case cut (upper) and vertical cut (lower).
(Figure 1).
The direct numerical simulations (DNSs) showed thatthe turbulent transport represented by the turbulent massand turbulent internal-energy fluxes is much larger as awhole, and much localized in the near surface region in
the non-locally driven cases.[2] These properties ofturbulence and its transport could not be reproduced bythe usual turbulence model based on the gradientdiffusion approximation with mixing-length theory.
Temporal variation of turbulence properties along themean motions is one of the representative nonequilibrium effects. In the framework of the multiplescale direct-interaction approximation, where theresponse-function formulation is adopted, the nonequilibrium effect is incorporated through the advectiveor Lagrangian derivative of the velocity fluctuations. In
this framework, the length scale of turbulence isexpressed in terms ofand and their Lagrangianderivatives. With this non-equilibrium effect, forexample, the eddy viscosity is expressed as,(1)
whereis the equilibrium eddy viscosity.
In order to capture the plume motions, we adopt a time-space double averaging procedure, With this formulation,a field quantity is divided into three components: thespatial average of the time average, coherent

fluctuationas, and incoherent fluctuation(2)
The energy transfer between the coherent and incoherent
fluctuation energies are mediated by(3)
whereandare the gains of the incoherent andcoherent fluctuation energies due to the transfer.
Equation (3) shows that the shear of the coherent orCooling
plume motions coupled
Plumewith the incoherent or (Coherent fluctuation)r a n d o m fl u c t u a t i o n sdetermines the energy Random noise(Incoherent fluctuation)Ptransfer between the plumeand random fluctuations Figure 2: Interaction between thecoherent and incoherent fluctuations.
(Figure 2).
The turbulent internal energy flux is modeled asK"

(4)
whereis the ambient mean density,

constant, anddefined byis the modelis the non-equilibrium effect factor
(5)

Here,
is the equilibrium
turbulent internal-energy
diffusivity.
In Figure 3, the spatial
distributions of the turbulent

1-(z/d)1z<e'u' >

Figure 3: Spatial distribution ofinternal-energy flux
obtained from the DNSs and the turbulent internal-energy
the turbulence model with
the non-equilibrium effect are shown. We see that thenon-equilibrium model well reproduces the DNS result
for the non-locally driven convection, which cannot bereproduced by the standard gradient-diffusion modelwith the mixing-length theory.

References[1] N. Yokoi, Y. Masada, and T. Takiwaki, Mon. Not.
Roy. Astro. Soc., 516, 2718 (2022)[2] N. Yokoi, Atmosphere, 14, 1013 (2023)