arXiv subjects
Publications and source records attributed to Chuong V. Tran.
In this brief note we study the $n$-dimensional magnetohydrodynamic equations with hyper-viscosity and zero resistivity. We prove global regularity of solutions when the hyper-viscosity is sufficiently strong.
In this article we study the global regularity of 2D generalized magnetohydrodynamic equations (2D GMHD), in which the dissipation terms are $- ν(- \triangle)^α u$ and $- κ(-\triangle)^β b$. We show that smooth solutions are global in the following three cases: $α\geqslant 1 / 2, β\geqslant 1$; $0 \leqslant α< 1 / 2, 2 α+ β> 2$; $α\geqslant 2, β= 0$. We also show that in the inviscid case $ν= 0$, if $β> 1$, then smooth solutions are global as long as the direction of the magnetic field remains smooth enough.
In Kolmogorov's phenomenological theory of turbulence, the energy spectrum in the inertial range scales with the wave number $k$ as $k^{-5/3}$ and extends up to a dissipation wave number $k_ν$, which is given in terms of the energy dissipation rate $ε$ and viscosity $ν$ by $k_ν\propto(ε/ν^3)^{1/4}$. This result leads to Landau's heuristic estimate for the number of degrees of freedom that scales as $\Re^{9/4}$, where $\Re$ is the Reynolds number. Here we consider the possibility of establishing a quantitative basis for these results from first principles. In particular, we examine the extent to which they can be derived from the three-dimensional Navier--Stokes system, making use of Kolmogorov's hypothesis of finite and viscosity-independent energy dissipation only. It is found that the Taylor microscale wave number $k_T$ (a close cousin of $k_ν$) can be expressed in the form $k_T \le CU/ν= (CU/\norm{\u})^{1/2}(ε/ν^3)^{1/4}$. Here $U$ and $\norm{\u}$ are, respectively, a ``microscale'' velocity and the root mean square velocity, and $C\le1$ is a dynamical parameter. This result can be seen to be in line with Kolmogorov's prediction for $k_ν$. Furthermore, it is shown that the minimum number of greatest Lyapunov exponents whose sum becomes negative does not exceed $\Re^{9/4}$, where $\Re$ is defined in terms of an average energy dissipation rate, the system length scale, and $ν$. This result is in a remarkable agreement with the Landau estimate, up to a presumably slight discrepancy between the conventional and the present energy dissipation rates used in the definition of $\Re$.
Travelling-wave solutions of the inviscid Burgers equation having smooth initial wave profiles of suitable shapes are known to develop shocks (infinite gradients) in finite times. Such singular solutions are characterized by energy spectra that scale with the wave number $k$ as $k^{-2}$. **** In this study, we carry out an analysis which verifies the dynamical features described above and derive upper bounds for $ε$ and $N$. It is found that $ε$ satisfies $ε\le ν^{2α-1}\norm{u_*}_\infty^{2(1-α)} \norm{(-Δ)^{α/2}u_*}^2$, where $α<1$ and $u_*=u(x,t_*)$ is the velocity field at $t=t_*$. Given $ε>0$ in the limit $ν\to0$, this implies that the energy spectrum remains no steeper than $k^{-2}$ in that limit. For the critical $k^{-2}$ scaling, the bound for $ε$ reduces to $ε\le\sqrt{3}k_0\norm{u_0}_\infty\norm{u_0}^2$, where $k_0$ marks the lower end of the inertial range and $u_0=u(x,0)$. This implies $N\le\sqrt{3}L\norm{u_0}_\infty/ν$, where $L$ is the domain size, which is shown to coincide with a rigorous estimate for the number of degrees of freedom defined in terms of local Lyapunov exponents. We demonstrate both analytically and numerically an instance where the $k^{-2}$ scaling is uniquely realizable. The numerics also return $ε$ and $t_*$, consistent with analytic values derived from the corresponding limiting weak solution.
We study the small-scale behavior of generalized two-dimensional turbulence governed by a family of model equations, in which the active scalar $θ=(-Δ)^{α/2}ψ$ is advected by the incompressible flow $\u=(-ψ_y,ψ_x)$. The dynamics of this family are characterized by the material conservation of $θ$, whose variance $<θ^2>$ is preferentially transferred to high wave numbers. As this transfer proceeds to ever-smaller scales, the gradient $\nablaθ$ grows without bound. This growth is due to the stretching term $(\nablaθ\cdot\nabla)\u$ whose ``effective degree of nonlinearity'' differs from one member of the family to another. This degree depends on the relation between the advecting flow $\u$ and the active scalar $θ$ and is wide ranging, from approximately linear to highly superlinear. Linear dynamics are realized when $\nabla\u$ is a quantity of no smaller scales than $θ$, so that it is insensitive to the direct transfer of the variance of $θ$, which is nearly passively advected. This case corresponds to $α\ge2$, for which the growth of $\nablaθ$ is approximately exponential in time and non-accelerated. For $α<2$, superlinear dynamics are realized as the direct transfer of $<θ^2>$ entails a growth in $\nabla\u$, thereby enhancing the production of $\nablaθ$. This superlinearity reaches the familiar quadratic nonlinearity of three-dimensional turbulence at $α=1$ and surpasses that for $α<1$. The usual vorticity equation ($α=2$) is the border line, where $\nabla\u$ and $θ$ are of the same scale, separating the linear and nonlinear regimes of the small-scale dynamics. We discuss these regimes in detail, with an emphasis on the locality of the direct transfer.
We derive upper bounds for the number of degrees of freedom of two-dimensional Navier--Stokes turbulence freely decaying from a smooth initial vorticity field $ω(x,y,0)=ω_0$. This number, denoted by $N$, is defined as the minimum dimension such that for $n\ge N$, arbitrary $n$-dimensional balls in phase space centred on the solution trajectory $ω(x,y,t)$, for $t>0$, contract under the dynamics of the system linearized about $ω(x,y,t)$. In other words, $N$ is the minimum number of greatest Lyapunov exponents whose sum becomes negative. It is found that $N\le C_1R_e$ when the phase space is endowed with the energy norm, and $N\le C_2R_e(1+\ln R_e)^{1/3}$ when the phase space is endowed with the enstrophy norm. Here $C_1$ and $C_2$ are constant and $R_e$ is the Reynolds number defined in terms of $ω_0$, the system length scale, and the viscosity $ν$. The linear (or nearly linear) dependence of $N$ on $R_e$ is consistent with the estimate for the number of active modes deduced from a recent mathematical bound for the viscous dissipation wave number. This result is in a sharp contrast to the forced case, for which well-known estimates for the Hausdorff dimension $D_H$ of the global attractor scale highly superlinearly with $ν^{-1}$. We argue that the "extra" dependence of $D_H$ on $ν^{-1}$ is not an intrinsic property of the turbulent dynamics. Rather, it is a "removable artifact," brought about by the use of a time-independent forcing as a model for energy and enstrophy injection that drives the turbulence.
This study revisits the problem of advective transfer and spectra of a diffusive scalar field in large-scale incompressible flows in the presence of a (large-scale) source. By ``large-scale'' it is meant that the spectral support of the flows is confined to the wave-number region $k k_d$ is bounded from above by $Uk_dkΘ(k,t)$, where $U$ denotes the maximum fluid velocity and $Θ(k,t)$ is the spectrum of the scalar variance, defined as its average over the shell $(k-k_d,k+k_d)$. For a given flux, say $\vartheta>0$, across $k>k_d$, this bound requires $$Θ(k,t)\ge \frac{\vartheta}{Uk_d}k^{-1}.$$ This is consistent with recent numerical studies and with Batchelor's theory that predicts a $k^{-1}$ spectrum (with a slightly different proportionality constant) for the viscous-convective range, which could be identified with $(k_d,k_κ)$. Thus, Batchelor's formula for the variance spectrum is recovered by the present method in the form of a critical lower bound. The present result applies to a broad range of large-scale advection problems in space dimensions $\ge2$, including some filter models of turbulence, for which the turbulent velocity field is advected by a smoothed version of itself. For this case, $Θ(k,t)$ and $\vartheta$ are the kinetic energy spectrum and flux, respectively.
This study is concerned with the decay behaviour of a passive scalar $θ$ in three-dimensional flows having bounded velocity gradients. Given an initially smooth scalar distribution, the decay rate $d<θ^2>/dt$ of the scalar variance $<θ^2>$ is found to be bounded in terms of controlled physical parameters. Furthermore, in the zero diffusivity limit, $κ\to0$, this rate vanishes as $κ^{α_0}$ if there exists an $α_0\in(0,1]$ independent of $κ$ such that $<|(-Δ)^{α/2}θ|^2><\infty$ for $α\leα_0$. This condition is satisfied if in the limit $κ\to0$, the variance spectrum $Θ(k)$ remains steeper than $k^{-1}$ for large wave numbers $k$. When no such positive $α_0$ exists, the scalar field may be said to become virtually singular. A plausible scenario consistent with Batchelor's theory is that $Θ(k)$ becomes increasingly shallower for smaller $κ$, approaching the Batchelor scaling $k^{-1}$ in the limit $κ\to0$. For this classical case, the decay rate also vanishes, albeit more slowly -- like $(\ln P_r)^{-1}$, where $P_r$ is the Prandtl or Schmidt number. Hence, diffusion anomaly is ruled out for a broad range of scalar distribution, including power-law spectra no shallower than $k^{-1}$. The implication is that in order to have a $κ$-independent and non-vanishing decay rate, the variance at small scales must necessarily be greater than that allowed by the Batchelor spectrum. These results are discussed in the light of existing literature on the asymptotic exponential decay $<θ^2>\sim e^{-γt}$, where $γ>0$ is independent of $κ$.
This study is concerned with the diffusion of a passive scalar $Θ(\r,t)$ advected by general $n$-dimensional shear flows $\u=u(y,z,...,t)\hat{x}$ having finite mean-square velocity gradients. The unidirectionality of the incompressible flows conserves the stream-wise scalar gradient, $\partial_xΘ$, allowing only the cross-stream components to be amplified by shearing effects. This amplification is relatively weak because an important contributing factor, $\partial_xΘ$, is conserved, effectively rendering a slow diffusion process. It is found that the decay of the scalar variance $<Θ^2>$ satisfies $d<Θ^2>/dt\ge -Cκ^{1/3}$, where $C>0$ is a constant, depending on the fluid velocity gradients and initial distribution of $Θ$, and $κ$ is the molecular diffusivity. This result generalizes to axisymmetric flows on the plane and on the sphere having finite mean-square angular velocity gradients.
The behaviour of turbulent flows within the single-layer quasi-geostrophic (Charney--Hasegawa--Mima) model is shown to be strongly dependent on the Rossby deformation wavenumber $λ$ (or free-surface elasticity). Herein, we derive a bound on the inverse energy transfer, specifically on the growth rate $\d\ell/\dt$ of the characteristic length scale $\ell$ representing the energy centroid. It is found that $\d\ell/\dt\le2\norm q_\infty/(\ell_sλ^2)$, where $\norm q_\infty$ is the supremum of the potential vorticity and $\ell_s$ represents the potential enstrophy centroid of the reservoir, both invariant. This result implies that in the potential energy dominated regime ($\ell\ge\ell_s\ggλ^{-1}$), the inverse energy transfer is strongly impeded, in the sense that under the usual time scale no significant transfer of energy to larger scales occurs. The physical implication is that the elasticity of the free surface impedes turbulent energy transfer in wavenumber space, effectively rendering large-scale vortices long-lived and inactive. Results from numerical simulations of forced-dissipative turbulence confirm this prediction.
Freely decaying two-dimensional Navier--Stokes turbulence is studied. The conservation of vorticity by advective nonlinearities renders a class of Casimirs that decays under viscous effects. A rigorous constraint on the palinstrophy production by nonlinear transfer is derived, and an upper bound for the enstrophy dissipation is obtained. This bound depends only on the decaying Casimirs, thus allowing the enstrophy dissipation to be bounded from above in terms of initial data of the flows. An upper bound for the enstrophy dissipation wavenumber is derived and the new result is compared with the classical dissipation wavenumber.
The inverse transfer in two-dimensional turbulence governed by the surface quasi-geostrophic (SQG) equation is studied. The nonlinear transfer of this system conserves the two quadratic quantities $Ψ_1=<|(-Δ)^{1/4}ψ|^2>/2$ and $Ψ_2=<|(-Δ)^{1/2}ψ|^2>/2$ (kinetic energy), where $ψ$ is the streamfunction and $<\cdot>$ denotes a spatial average. In the limit of infinite domain, the kinetic energy density $Ψ_2$ remains bounded. For power-law inverse-transfer region, the inverse flux of $Ψ_1$ diminishes as it proceeds toward sufficiently low wavenumbers, implying that no persistent inverse cascade of $Ψ_1$ is sustainable. The unrealizability of an inverse cascade of $Ψ_1$ implies that there is no direct cascade of $Ψ_2$. Hence, the dual-cascade picture which is widely believed to be realizable in two-dimensional Navier--Stokes turbulence does not apply to SQG turbulence. Numerical results supporting the theoretical predictions are presented.
We study the inverse energy transfer in forced two-dimensional (2D) Navier--Stokes turbulence in a doubly periodic domain. It is shown that an inverse energy cascade that carries a nonzero fraction of the injected energy to the large scales via a power-law energy spectrum $\propto k^{-α}$ requires that $α\ge5/3$. This result is consistent with the classical theory of 2D turbulence that predicts a $k^{-5/3}$ inverse-cascading range, thus providing for the first time a rigorous basis for this important feature of the theory. We derive bounds for the Kolmogorov constant $C$ in the classical energy spectrum $E(k)=Cε^{2/3}k^{-5/3}$, where $ε$ is the energy injection rate. Issues related to Kraichnan's conjecture of energy condensation and to power-law spectra as the quasi-steady dynamics become steady are discussed.
The large-scale energy spectrum in two-dimensional turbulence governed by the surface quasi-geostrophic (SQG) equation $$\partial_t(-Δ)^{1/2}ψ+J(ψ,(-Δ)^{1/2}ψ) =μΔψ+f$$ is studied. The nonlinear transfer of this system conserves the two quadratic quantities $Ψ_1=<[(-Δ)^{1/4}ψ]^2>/2$ and $Ψ_2=<[(-Δ)^{1/2}ψ]^2>/2$ (kinetic energy), where $<\cdot>$ denotes a spatial average. The energy density $Ψ_2$ is bounded and its spectrum $Ψ_2(k)$ is shallower than $k^{-1}$ in the inverse-transfer range. For bounded turbulence, $Ψ_2(k)$ in the low-wavenumber region can be bounded by $Ck$ where $C$ is a constant independent of $k$ but dependent on the domain size. Results from numerical simulations confirming the theoretical predictions are presented.
Two-dimensional turbulence governed by the so-called $α$ turbulence equations, which include the surface quasi-geostrophic equation ($α=1$), the Navier--Stokes system ($α=2$), and the governing equation for a shallow flow on a rotating domain driven by a uniform internal heating ($α=3$), is studied here in both the unbounded and doubly periodic domains. This family of equations conserves two inviscid invariants (energy and enstrophy in the Navier--Stokes case), the dynamics of which are believed to undergo a dual cascade. It is shown that an inverse cascade can exist in the absence of a direct cascade and that the latter is possible only when the inverse transfer rate of the inverse-cascading quantity approaches its own injection rate. Constraints on the spectral exponents in the wavenumber ranges lower and higher than the injection range are derived. For Navier--Stokes turbulence with moderate Reynolds numbers, the realization of an inverse energy cascade in the complete absence of a direct enstrophy cascade is confirmed by numerical simulations.
This study is concerned with how the attractor dimension of the two-dimensional Navier--Stokes equations depends on characteristic length scales, including the system integral length scale, the forcing length scale, and the dissipation length scale. Upper bounds on the attractor dimension derived by Constantin--Foias--Temam are analysed. It is shown that the optimal attractor-dimension estimate grows linearly with the domain area (suggestive of extensive chaos), for a sufficiently large domain, if the kinematic viscosity and the amplitude and length scale of the forcing are held fixed. For sufficiently small domain area, a slightly ``super-extensive'' estimate becomes optimal. In the extensive regime, the attractor-dimension estimate is given by the ratio of the domain area to the square of the dissipation length scale defined, on physical grounds, in terms of the average rate of shear. This dissipation length scale (which is not necessarily the scale at which the energy or enstrophy dissipation takes place) can be identified with the dimension correlation length scale, the square of which is interpreted, according to the concept of extensive chaos, as the area of a subsystem with one degree of freedom. Furthermore, these length scales can be identified with a ``minimum length scale'' of the flow, which is rigorously deduced from the concept of determining nodes.
We study the dual cascade scenario for two-dimensional turbulence driven by a spectrally localized forcing applied over a finite wavenumber range $[k_\min,k_\max]$ (with $k_\min > 0$) such that the respective energy and enstrophy injection rates $ε$ and $η$ satisfy $k_\min^2ε\leη\le k_\max^2ε$. The classical Kraichnan--Leith--Batchelor paradigm, based on the simultaneous conservation of energy and enstrophy and the scale-selectivity of the molecular viscosity, requires that the domain be unbounded in both directions. For two-dimensional turbulence either in a doubly periodic domain or in an unbounded channel with a periodic boundary condition in the across-channel direction, a direct enstrophy cascade is not possible. In the usual case where the forcing wavenumber is no greater than the geometric mean of the integral and dissipation wavenumbers, constant spectral slopes must satisfy $β>5$ and $α+β\ge8$, where $-α$ ($-β$) is the asymptotic slope of the range of wavenumbers lower (higher) than the forcing wavenumber. The influence of a large-scale dissipation on the realizability of a dual cascade is analyzed. We discuss the consequences for numerical simulations attempting to mimic the classical unbounded picture in a bounded domain.
We study two-dimensional turbulence in a doubly periodic domain driven by a monoscale-like forcing and damped by various dissipation mechanisms of the form $ν_μ(-Δ)^μ$. By ``monoscale-like'' we mean that the forcing is applied over a finite range of wavenumbers $k_\min \leq k \leq k_\max$, and that the ratio of enstrophy injection $η\geq 0$ to energy injection $ε\geq 0$ is bounded by $k_\min^2 ε\leq η\leq k_\max^2 ε$. It is shown that for $μ\geq 0$ the asymptotic behaviour satisfies (eqnarray) \norm u_1^2&\leq&k_\max^2\norm u^2,(eqnarray) where $\norm u^2$ and $\norm u_1^2$ are the energy and enstrophy, respectively. It is also shown that for Navier-Stokes turbulence ($μ= 1$), the time-mean enstrophy dissipation rate is bounded from above by $2ν_1 k_\max^2$. These results place strong constraints on the spectral distribution of energy and enstrophy and of their dissipation, and thereby on the existence of energy and enstrophy cascades, in such systems. In particular, the classical dual cascade picture is shown to be invalid for forced two-dimensional Navier--Stokes turbulence ($μ=1$) when it is forced in this manner. Inclusion of Ekman drag ($μ=0$) along with molecular viscosity permits a dual cascade, but is incompatible with the log-modified -3 power law for the energy spectrum in the enstrophy-cascading inertial range. In order to achieve the latter, it is necessary to invoke an inverse viscosity ($μ<0$).