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 $epsilon$ and $N$. It is found that $epsilon$ satisfies $epsilon le u^{2alpha-1} orm{u_*}_infty^{2(1-alpha)} orm{(-Delta)^{alpha/2}u_*}^2$, where $alpha<1$ and $u_*=u(x,t_*)$ is the velocity field at $t=t_*$. Given $epsilon>0$ in the limit $ uto0$, this implies that the energy spectrum remains no steeper than $k^{-2}$ in that limit. For the critical $k^{-2}$ scaling, the bound for $epsilon$ reduces to $epsilonlesqrt{3}k_0 orm{u_0}_infty orm{u_0}^2$, where $k_0$ marks the lower end of the inertial range and $u_0=u(x,0)$. This implies $Nlesqrt{3}L orm{u_0}_infty/ u$, 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 $epsilon$ 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 $theta=(-Delta)^{alpha/2}psi$ is advected by the incompressible flow $u=(-psi_y,psi_x)$. The dynamics of t
his family are characterized by the material conservation of $theta$, whose variance $<theta^2>$ is preferentially transferred to high wave numbers. As this transfer proceeds to ever-smaller scales, the gradient $ ablatheta$ grows without bound. This growth is due to the stretching term $( ablathetacdot abla)u$ 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 $theta$ and is wide ranging, from approximately linear to highly superlinear. Linear dynamics are realized when $ ablau$ is a quantity of no smaller scales than $theta$, so that it is insensitive to the direct transfer of the variance of $theta$, which is nearly passively advected. This case corresponds to $alphage2$, for which the growth of $ ablatheta$ is approximately exponential in time and non-accelerated. For $alpha<2$, superlinear dynamics are realized as the direct transfer of $<theta^2>$ entails a growth in $ ablau$, thereby enhancing the production of $ ablatheta$. This superlinearity reaches the familiar quadratic nonlinearity of three-dimensional turbulence at $alpha=1$ and surpasses that for $alpha<1$. The usual vorticity equation ($alpha=2$) is the border line, where $ ablau$ and $theta$ 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.
