On inertial-range scaling laws

Inertial-range scaling laws for two- and three-dimensional turbulence are re-examined within a unified framework. A new correction to Kolmogorov's $k^{-5/3}$ scaling is derived for the energy inertial range. A related modification is found to Kraichnan's logarithmically corrected two-dimensional enstrophy-range law that removes its unexpected divergence at the injection wavenumber. The significance of these corrections is illustrated with steady-state energy spectra from recent high-resolution closure computations. Implications for conventional numerical simulations are discussed. These results underscore the asymptotic nature of inertial-range scaling laws.Comment: 16 pages, postscript (uncompressed, not encoded

Large-scale energy spectra in surface quasi-geostrophic turbulence

The large-scale energy spectrum in two-dimensional turbulence governed by the surface quasi-geostrophic (SQG) equation $$\partial_t(-\Delta)^{1/2}\psi+J(\psi,(-\Delta)^{1/2}\psi) =\mu\Delta\psi+f$$ is studied. The nonlinear transfer of this system conserves the two quadratic quantities $\Psi_1=/2$ and $\Psi_2=/2$ (kinetic energy), where $$ denotes a spatial average. The energy density $\Psi_2$ is bounded and its spectrum $\Psi_2(k)$ is shallower than $k^{-1}$ in the inverse-transfer range. For bounded turbulence, $\Psi_2(k)$ 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.Comment: 11 pages, 4 figures, to appear in JF

Exactly Conservative Integrators

Traditional numerical discretizations of conservative systems generically yield an artificial secular drift of any nonlinear invariants. In this work we present an explicit nontraditional algorithm that exactly conserves these invariants. We illustrate the general method by applying it to the three-wave truncation of the Euler equations, the Lotka--Volterra predator--prey model, and the Kepler problem. This method is discussed in the context of symplectic (phase space conserving) integration methods as well as nonsymplectic conservative methods. We comment on the application of our method to general conservative systems.Comment: 30 pages, postscript (1.3MB). Submitted to SIAM J. Sci. Comput

Robust Exponential Runge-Kutta Embedded Pairs

Exponential integrators are explicit methods for solving ordinary differential equations that treat linear behaviour exactly. The stiff-order conditions for exponential integrators derived in a Banach space framework by Hochbruck and Ostermann are solved symbolically by expressing the Runge--Kutta weights as unknown linear combinations of phi functions. Of particular interest are embedded exponential pairs that efficiently generate both a high- and low-order estimate, allowing for dynamic adjustment of the time step. A key requirement is that the pair be robust: if the nonlinear source function has nonzero total time derivatives, the order of the low-order estimate should never exceed its design value. Robust exponential Runge--Kutta (3,2) and (4,3) embedded pairs that are well-suited to initial value problems with a dominant linearity are constructed.Comment: 24 pages, 8 figures. The Mathematica scripts mentioned in the paper can be found in: https://github.com/stiffode/expint

Nonlinear symmetric stability of planetary atmospheres

The energy–Casimir method is applied to the problem of symmetric stability in the context of a compressible, hydrostatic planetary atmosphere with a general equation of state. Formal stability criteria for symmetric disturbances to a zonally symmetric baroclinic flow are obtained. In the special case of a perfect gas the results of Stevens (1983) are recovered. Finite-amplitude stability conditions are also obtained that provide an upper bound on a certain positive-definite measure of disturbance amplitude

Case History of Seismic Base Isolation of a Building –The Foothill Communities Law and Justice Center

The Foothill Communities Law and Justice Center, located in seismically active Southern California, is the first building in the United States to be base isolated for seismic resistance. Natural rubber isolators with layers of steel plates were used to make the fundamental period of vibration of the base isolated building about twice as long as that for a comparable conventional fixed base building. Most earthquake energy is present in the shorter period ranges, and at longer periods, a building should be subjected to less earthquake input; this will allow buildings to be designed more economically and increase the likelihood of less damage, both structural and non- structural. The experience of the Law and Justice Center after three small earthquakes suggest that the concept is not only feasible, but may be the wave of the future for what would be relatively short period buildings