“…The latter rate suggests the presence of a square root singularity on and an inverse square root singularity on its derivatives, i.e. on the velocities (Salem 1939; Raisbeck 1955). Figure 3.( a ) Decay of in log-scale for the case of a bump topography with and , , , and .…”
Section: Resultsmentioning
confidence: 99%
“…The latter rate suggests the presence of a square root singularity on φ † and an inverse square root singularity on its derivatives, i.e. on the velocities (Salem 1939;Raisbeck 1955).…”
We model linear, inviscid, internal tides generated by the interaction of a barotropic tide with one-dimensional topography. Starting from the body-forcing formulation of the hydrodynamic problem, we derive a coupled-mode system (CMS) using a local eigenfunction expansion of the stream function. For infinitesimal topography, we solve this CMS analytically, recovering the classical weak topography approximation (WTA) formula for the barotropic-to-baroclinic energy conversion rate. For arbitrary topographies, we solve this CMS numerically. The CMS enjoys faster convergence with respect to existing modal solutions and can be applied in the subcritical and supercritical regimes for both ridges and shelf profiles. We show that the non-uniform barotropic tide affects the baroclinic field locally over topographies with large slopes and we study the dependence of the radiated energy conversion rate on the criticality. We show that non-radiating or weakly radiating topographies are common in the subcritical regime. We also assess the region of validity of the WTA approximation for the commonly used Gaussian ridge and a compactly supported bump ridge studied here for the first time. Finally, we provide numerical evidence showing that in the strongly supercritical regime, the energy conversion rate for a ridge (respectively shelf) approaches the value obtained by the knife-edge (respectively step) topography.
“…The latter rate suggests the presence of a square root singularity on and an inverse square root singularity on its derivatives, i.e. on the velocities (Salem 1939; Raisbeck 1955). Figure 3.( a ) Decay of in log-scale for the case of a bump topography with and , , , and .…”
Section: Resultsmentioning
confidence: 99%
“…The latter rate suggests the presence of a square root singularity on φ † and an inverse square root singularity on its derivatives, i.e. on the velocities (Salem 1939;Raisbeck 1955).…”
We model linear, inviscid, internal tides generated by the interaction of a barotropic tide with one-dimensional topography. Starting from the body-forcing formulation of the hydrodynamic problem, we derive a coupled-mode system (CMS) using a local eigenfunction expansion of the stream function. For infinitesimal topography, we solve this CMS analytically, recovering the classical weak topography approximation (WTA) formula for the barotropic-to-baroclinic energy conversion rate. For arbitrary topographies, we solve this CMS numerically. The CMS enjoys faster convergence with respect to existing modal solutions and can be applied in the subcritical and supercritical regimes for both ridges and shelf profiles. We show that the non-uniform barotropic tide affects the baroclinic field locally over topographies with large slopes and we study the dependence of the radiated energy conversion rate on the criticality. We show that non-radiating or weakly radiating topographies are common in the subcritical regime. We also assess the region of validity of the WTA approximation for the commonly used Gaussian ridge and a compactly supported bump ridge studied here for the first time. Finally, we provide numerical evidence showing that in the strongly supercritical regime, the energy conversion rate for a ridge (respectively shelf) approaches the value obtained by the knife-edge (respectively step) topography.
“…Our changes in the transparency, modeled as Dirac functions, become a piece-wise constant function after integration. Since F(d) has discontinuities, the rate of decay in the k th -Fourier coefficient is proportional to 1 k [49]. As shown in Section 2.3.1, it can be seen by Eq.…”
Section: Decay Rate Of Coefficientsmentioning
confidence: 93%
“…Eq. 5shows the rate at which the Fourier terms decay when increasing m. The magnitude of the terms with coefficients a k and b k decays at a rate of 1 π k compared to a 0 [49]. Thus, terms for k > 5 have a magnitude of about 5% or less compared to a 0 , and have thus a small impact on the reconstructed signal.…”
Over the past decades, scientific visualization became a fundamental aspect of modern scientific data analysis. Across all data-intensive research fields, ranging from structural biology to cosmology, data sizes increase rapidly. Dealing with the growing large-scale data is one of the top research challenges of this century. For the visual exploratory data analysis, interactivity, a view-dependent visibility optimization and frame coherence are indispensable. In this work, we extend the recent decoupled opacity optimization framework to enable a navigation without occlusion of important features through large geometric data. By expressing the accumulation of importance and optical depth in Fourier basis, the computation, evaluation and rendering of optimized transparent geometry become not only order-independent, but also operate within a fixed memory bound. We study the quality of our Fourier approximation in terms of accuracy, memory requirements and efficiency for both the opacity computation, as well as the order-independent compositing. We apply the method to different point, line and surface data sets originating from various research fields, including meteorology, health science, astrophysics and organic chemistry.
“…where a 2π-periodicity is assumed, and the coefficients a n and b n are calculated using the well-known integral expressions 2 . It is analytically known 2 that there are two characteristic features of Fourier expansions 7 .…”
While resistors with memory, sometimes called memristive elements (such as ReRAM cells), are often studied under conditions of periodic driving, little attention has been paid to the Fourier features of their memory response (hysteresis). Here we demonstrate experimentally that the hysteresis of memristive systems can be unambiguously distinguished from the linear or non-linear response of systems without hysteresis by the values of certain Fourier series coefficients. We also show that the Fourier series convergence depends on driving conditions. These results may be used to quantify the memory content of resistive memories, and tune their Fourier spectrum according to the excitation signal.
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