On the stability and bifurcation of the non-rotating Boussinesq equation with the Kolmogorov forcing at a low Péclet number

Highlights

• It proves that a stratified shear flow governed by the non-rotating incompressible Boussinesq equation at a low Péclet number becomes unstable as the Reynold number Re is above a threshold.

• There exists a supercritical Hopf bifurcation in the non-rotating incompressible Boussinesq equation at the threshold.

• An upper boundedness of the threshold is derived.

• A stable periodic solution emerges in the non-rotating incompressible Boussinesq equation at the threshold, which describes an oscillating thermal convection in a highly stratified shear flow arising in the atmosphere or interior of many stellar systems.

Abstract

This study examines the stability and potential bifurcations of a stratified shear flow governed by the non-rotating incompressible Boussinesq equation at a low Péclet number. For the ratio of the vertical scale to the horizontal scale of a stratified flow $a\in [\sqrt{3}/4,\sqrt{3}/2),$ it is shown that there exists a threshold for the Reynold number Re above which the steady stratified shear flow driven by the Kolmogorov forcing becomes linearly unstable. As a result, the Boussinesq equation exhibits a Hopf bifurcation. To further determine the type of the Hopf bifurcation, the model is reduced to a low-order system whose numerical analyses reveal that the Hopf bifurcation is supercritical. That is, a stable periodic solution emerges, which describes an oscillating thermal convection in a highly stratified shear flow arising in the atmosphere or interior of many stellar systems with low Péclet numbers.

Introduction

The stability of stratified shear flows that is responsible for the onset of turbulence has attracted great attention in the fluid dynamics research [1]. The first linear stability analysis of a stratified shear flow in the limit of zero viscosity and diffusivity was due to Taylor [2], who examined both continuously and discretely varying stratification and shear profiles. Taylor’s results were subsequently generalized by Miles [3], [4] and Howard [5] to a larger class of fluid systems.

For thermally stratified flows, thermal diffusion can have a significant influence on the development of shear instabilities by damping the buoyancy restoring force. This effect was first considered by Townsend [6] in the study of atmospheric flows. He showed that the thermal adjustment of a fluid parcel to its surroundings, by radiative heating and cooling or by thermal conduction, always destabilizes the flow and increases the critical Richardson number for linear stability. For highly stratified flows such as those in stellar interiors where the Prandtl number is typically very small [7], thermally diffusive shear flows can exist even when viscosity is sufficiently small to prevent the development of instability [8]. This behavior indicates that the opposing effect of viscosity and thermal diffusion on the thermally stratified flows for those astrophysical systems cannot be neglected.

In a general form, the governing equation for the thermally-stratified flows with both the effects of viscosity and thermal diffusion included is given by the Boussinesq equation as follows [9], given by

$$\begin{array}{cc}& {\displaystyle \frac{\partial \mathbf{u}}{\partial t}}+\mathbf{u}·\nabla \mathbf{u}=-{\displaystyle \frac{1}{{\rho }_0}}\nabla p+\alpha gT{\mathbf{e}}_z+\nu \Delta \mathbf{u},\hfill \\ \\ & \nabla ·\mathbf{u}=0,\hfill \\ \\ & {\displaystyle \frac{\partial T}{\partial t}}+\mathbf{u}·\nabla T+w{T}_0={\kappa }_T\Delta T,\hfill \end{array}$$

where $\mathbf{u}=(u,v,w)$ is the fluid velocity, p is the pressure, T represents the temperature, g the gravity constant, e_z is the unit vector in the vertical direction, μ is the viscosity constant, κ_T is the temperature diffusion constant, ρ₀ is the mean density of the region considered, α is the coefficient of the thermal expansion, and T₀ is the constant background temperature gradient in the vertical direction.

Eq. (1) has been widely employed to study the thermal convection arising in the Earth’s atmosphere and stellar [see, e.g., [10], [11], and references therein]. Due to its important role in fluid dynamics, the Boussinesq equation and other variants [12], [13] of which for different problems have been extensively studied from different angles. For example, the existence of global attractor of Eq. (1) on a bounded domain was shown by Foias et al. work [14]. Likewise, [15] showed the existence of global attractor of a model with vanishing diffusion on a periodic domain, while Pellew and Southwell [16] first addressed the principle of exchange of stabilities condition for the Rayleigh-Bénard problem governed by the Eq.  (1). The bifurcation of Eq.  (1) associated with the Rayleigh-Bénard convection is also well-known, see e.g. [17], [18] for the linear stability analysis, and [19], [20] for nonlinear theories, among many others [21], [22]. In their recent studies, Ma and Wang [19] derived the dynamic transition of Eq. (1), and showed that Eq. (1) could bifurcate from the zero solution to a local attractor containing exactly eight singular points when the Rayleigh number crosses the first critical Rayleigh number, under some configuration of boundary conditions. These results have been then generalized by Han et al. [23], in which a heating term was included in the diffusion equation.

For thermally-stratified flows in many astrophysical systems, thermal diffusivity is generally very high, leading to steady temperature fluctuations associated purely with vertical motion. For this case, the corresponding quasi-static approximation can be assumed to study the low-Prandtl number thermal convection in these astrophysical systems [24], [25]. With the help of the quasi-static approximation, Lignières [26] showed that the standard Boussinesq Eq. (1) can be reduced to

$$\begin{array}{cc}& {\displaystyle \frac{\partial \mathbf{u}}{\partial t}}+\mathbf{u}·\nabla \mathbf{u}=-{\displaystyle \frac{1}{{\rho }_0}}\nabla p+{\displaystyle \frac{\alpha g{T}_0}{{\kappa }_T}}{\Delta }^{-1}w{\mathbf{e}}_z+\nu \Delta \mathbf{u},\hfill \\ \\ & \nabla ·\mathbf{u}=0,\hfill \end{array}$$

which is valid in the asymptotic low-Péclet-number limit. Here ${\Delta }^{-1}$ denotes the inverse of Laplacian whose range follows the boundary conditions for the T variable. Recently, Garaud et al. [27] studied the energy stability properties of stably stratified shear flows with a positive background temperature gradient T₀ in the vertical direction, where the shear is explicitly determined by the reduced model (2) in presence of an external forcing term F/ρ₀ where the body force is given by

$$\mathbf{F}=F_0\sin \left(k\pi z\right){\mathbf{e}}_{\mathbf{x}}.$$

They extended Squire’s theorem to the reduced model (2), and showed that the first unstable mode is always two-dimensional. An energy stability analysis of the reduced model (2) was also performed, which showed that for a given value of the Reynolds number above a critical strength of the stratification, any smooth periodic shear flow is stable to perturbations of arbitrary amplitudes. However, the linear instability of the shear flow driven by different body-force is still open.

Motivated by the Garaud et al. [27], we aim to investigate in this study the stability of thermal shear flows with a negative background temperature gradient and low-Péclet-number from the perspective of dynamic transition and bifurcation introduced in Ma and Wang [28], whose reduced model now takes the form of

$$\begin{array}{cc}& {\displaystyle \frac{\partial \mathbf{u}}{\partial t}}+\mathbf{u}·\nabla \mathbf{u}=-{\displaystyle \frac{1}{{\rho }_0}}\nabla p-{\displaystyle \frac{\alpha g{T}_0}{{\kappa }_T}}{\Delta }^{-1}w{\mathbf{e}}_z+\nu \Delta \mathbf{u}+{\displaystyle \frac{1}{{\rho }_0}}\mathbf{F},\hfill \\ \\ & \nabla ·\mathbf{u}=0.\hfill \end{array}$$

This model is defined on the vertical plane (x, z) ∈ [0, 2L] × [0, 2H] where L and H are the horizontal and vertical dimensions of the model domain. Note here that T₀ denotes the absolute value of the temperature gradient. The forcing F of interest in this work is assumed to be a Kolmogorov forcing of the form

$$\mathbf{F}={\pi }^2{F}_0\cos \left({\displaystyle \frac{\pi z}{H}}\right){\mathbf{e}}_{\mathbf{x}},$$

which has been widely employed in many different problems [29], [30], [31], [32], [33], [34]. It is apparent that the above Kolmogorov forcing results in a viscous Kolmogorov flow (a stratified shear flow) as follows

$${\mathbf{u}}_{\mathbf{s}}={\displaystyle \frac{F_0{H}^2}{{\rho }_0\nu }}\cos \left({\displaystyle \frac{\pi z}{H}}\right){\mathbf{e}}_{\mathbf{x}},$$

which is different from the mean flow examined in Garaud et al. [27].

In this study, the linear stability and transition of the aforementioned steady shear flow will be studied by employing the method developed by Chen et al. [35], Chen and Price [36]. Our main results are summarized as follows:

• For the aspect ratio $H/L=a\in [\sqrt{3}/4,\sqrt{3}/2),$ there exists a threshold Re_c of the Reynolds number above which the steady shear flow becomes linearly unstable. In addition, we find that $R{e}_c<\frac{{\pi }^4{(a^2+1/4)}^3{\kappa }_T{F}_0}{a^2\alpha T_0{\rho }_0\nu }.$

• There exists a Hopf bifurcation in the system (3) at the critical threshold Re_c. That is, a periodic solution is bifurcated from the Kolmogorov flow.

• Numerical analyses for the low-order system in the vicinity of the threshold show that the Hopf bifurcation is supercritical and the corresponding dynamic transition is type-1 transition (a continuous transition. For types of transitions see [28]). In other words, the bifurcated periodic solution is stable.

• The stable periodic solution describes the coherent large-scale periodic convection in astrophysical systems whose vertical temperature gradient is negative such as the Sun’s interior.

The rest of this work will be organized as follows. The non-dimensional perturbation equations are given in Section 2. In Section 3, we perform the spectrum analysis for the linear part of perturbation equations, and the linear instability of the shear flow is shown. In Section 4, we derive the reduced low-order system by which a theorem involved the Hopf bifurcation and transition associated with the shear flow is established. Numerical results are arranged in Section 5, and concluding remarks are given in the final section.