Three dimensional roll-type double-diffusive convection in a horizontally infinite layer of an uncompressible liquid is considered in the neighborhood of Hopf bifurcation points. A system of amplitude equations for the variations of convective rolls amplitude is derived by multiple-scaled method. An attention is paid to an interaction of convection and horizontal vortex. Different cases of the derived equations are discussed.
Deep Dive into Amplitude equations for 3D double-diffusive convection interacted with a horizontal vortex.
Three dimensional roll-type double-diffusive convection in a horizontally infinite layer of an uncompressible liquid is considered in the neighborhood of Hopf bifurcation points. A system of amplitude equations for the variations of convective rolls amplitude is derived by multiple-scaled method. An attention is paid to an interaction of convection and horizontal vortex. Different cases of the derived equations are discussed.
Physical systems, in which double-diffusion induced convection plays an essential role, are often found in nature. There are two components with significantly different diffusion coefficients in such systems. It can be heat and salt in the sea water, heat and helium in stellar atmospheres, or two reagents in chemical reactors. As a result of various spatial distribution of these components in a gravitational field arises convection, which can have various forms and lead to a variety of phenomena [1]. Widely known, for example, salt fingers are, arising in salted and warmed from above water. It is understandable that the results of double-diffusive convection, for example, in the ocean, can be applied to double-diffusive convection in astrophysical systems or in chemical reactor.
There are a number of works devoted to various theoretical models of systems with double-diffusive convection. In 80-90 years the formation of structures in the neighborhood of Hopf bifurcation points for the horizontally translation-invariant systems was actively studied in some works. The development of oscillations in such systems give rise to different types of waves (eg, standing, running, modulated, chaotic), which well described by a generalized Ginzburg-Landau equations [3,5]. The equations of this type must be derived from the basic system of partial differential equations for the given physical system by asymptotic methods. However, a full and well-grounded derivation of amplitude equations for systems with double-diffusive convection (especially threedimensional) is still poorly represented in the literature.
The purpose of this work is the derivation of amplitude equations for the three-dimensional double-diffusive system in the neighborhood of Hopf bifurcation points for the case of roll-type convection. This extends the idea of previous work [3,4], where a two-dimensional and threedimensional convection in a square-cells was investigated by alike methods. * Electronic address: skozi@poi.dvo.ru
Consider 3D double-diffusive convection in a liquid layer of a width h, confined by two plane horizontal boundaries. The liquid layer is heated and salted from below. The governing equations in this case are hydrodynamical equations for a liquid mixture in the gravitational field [6]:
Where v(t, x, y, z) is the velocity field of liquid, T (t, x, y, z) is the temperature, S(t, x, y, z) is the salt concentration, p(t, x, y, z) is the pressure, ρ(t, x, y, z) is the density of liquid, g is the acceleration of gravity, ν is the kinematic viscosity of fluid, χ is the thermal diffusivity of the liquid, D is the salt diffusivity. Cartesian frame with the horizontal x-axis and y-axis is used, while the z-axis is directed upward and t is the time variable. Distributed sources of heat and salt are absent. On the upper and lower boundaries of the layer the constant values of temperature and salinity are supported, higher at the lower boundary. The governing equations are transformed into dimensionless form with the use of Boussinesq approximation and following units for length, time, velocity, pressure, temperature and salinity respectively: h, h 2 /χ, χ/h, ρ 0 χ 2 /h 2 , T ∆ , S ∆ , where T ∆ and S ∆ are temperature and salinity differences across the layer.
The dimensionless governing equations for momentum and diffusion of temperature and salt are [4]:
Where σ = ν/χ is the Prandtl number (σ ≈ 7.0), τ = D/χ is the Lewis number (0 < τ < 1, usually τ = 0.01 -0.1). R T = (gα ′ h 3 /χν)T ∆ is the temperature Rayleigh number and R S = (gγ ′ h 3 /χν)S ∆ is the salinity Rayleigh number.
Free-slip boundary conditions are used for the dependent variables (the horizontal velocity component is undefined):
It is believed that they are suitable to describe the convection in the inner layers of liquid and do not change significantly the convective instability occurrence criteria for the investigated class of systems [7].
Consider the equations for double-diffusive convection in the vicinity of a bifurcation point, the temperature and salinity Rayleigh numbers for which are designated as R * T and R * S respectively. In this case the Rayleigh numbers can be represented as follows:
Values of r T and r S are of unit order, and the small parameter ε shows how far from the bifurcation point the system is. To derive the amplitude equations we use the derivative-expansion method, which is a variant of multiple-scale method [8,9]. Introduce the slow variables:
In accordance with the method chosen, we assume that the dependent variables now depend on t, T 1 , T 2 , x, y, z, X 1 , Y 1 , which are considered independent. Also replace the derivatives in equations ( 1) for the prolonged ones by the rules:
Then the equations ( 1) can be written as:
We seek solutions of these equations in the form of asymptotic series in powers of small parameter ε:
After their substitution in (2) and collection the terms at ε n we obtain the systems of equations to dete
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