Spectral methods in time for hyperbolic equations

A pseudospectral numerical scheme for solving linear, periodic, hyperbolic problems is described. It has infinite accuracy both in time and in space. The high accuracy in time is achieved without increasing the computational work and memory space which is needed for a regular, one step explicit scheme. The algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution. The class of algorithms referred to consists of all explicit schemes which may be represented as a polynomial in the spatial operator

High degree interpolation polynomial in Newton form

Polynomial interpolation is an essential subject in numerical analysis. Dealing with a real interval, it is well known that even if f(x) is an analytic function, interpolating at equally spaced points can diverge. On the other hand, interpolating at the zeroes of the corresponding Chebyshev polynomial will converge. Using the Newton formula, this result of convergence is true only on the theoretical level. It is shown that the algorithm which computes the divided differences is numerically stable only if: (1) the interpolating points are arranged in a different order, and (2) the size of the interval is 4

Polynomial approximation of functions of matrices and its application to the solution of a general system of linear equations

During the process of solving a mathematical model numerically, there is often a need to operate on a vector v by an operator which can be expressed as f(A) while A is NxN matrix (ex: exp(A), sin(A), A sup -1). Except for very simple matrices, it is impractical to construct the matrix f(A) explicitly. Usually an approximation to it is used. In the present research, an algorithm is developed which uses a polynomial approximation to f(A). It is reduced to a problem of approximating f(z) by a polynomial in z while z belongs to the domain D in the complex plane which includes all the eigenvalues of A. This problem of approximation is approached by interpolating the function f(z) in a certain set of points which is known to have some maximal properties. The approximation thus achieved is almost best. Implementing the algorithm to some practical problem is described. Since a solution to a linear system Ax = b is x= A sup -1 b, an iterative solution to it can be regarded as a polynomial approximation to f(A) = A sup -1. Implementing the algorithm in this case is also described

The eigenvalues of the pseudospectral Fourier approximation to the operator sin (2x) d/dx

It is shown that the eigenvalues Z sub i of the pseudospectral Fourier approximation to the operator sin(2x) curly d/curly dx satisfy (R sub e) (Z sub i) = + or - 1 or (R sub e)(Z sub I) = 0. Whereas this does not prove stability for the Fourier method, applied to the hyperbolic equation U sub t = sin (2x)(U sub x) - pi x pi; it indicates that the growth in time of the numerical solution is essentially the same as that of the solution to the differential equation

Sea Level Rise, Spatially Uneven and Temporally Unsteady: Why the U.S. East Coast, the Global Tide Gauge Record, and the Global Altimeter Data Show Different Trends

Impacts of ocean dynamics on spatial and temporal variations in sea level rise (SLR) along the U.S. East Coast are characterized by empirical mode decomposition analysis and compared with global SLR. The findings show a striking latitudinal SLR pattern. Sea level acceleration consistent with a weakening Gulf Stream is maximum just north of Cape Hatteras and decreasing northward, while SLR driven by multidecadal variations, possibly from climatic variations in subpolar regions, is maximum in the north and decreasing southward. The combined impact of sea level acceleration and multidecadal variations explains why the global mean SLR obtained from similar to 20 years of altimeter data is about twice the century-long global SLR obtained from tide gauge data. The sea level difference between Bermuda and the U.S. coast is highly correlated with the transport of the Atlantic Overturning Circulation, a result with implications for detecting past and future climatic changes using tide gauge data

Decadal Variabilities of the Upper Layers of the Subtropical North Atlantic: An Ocean Model Study

Numerical simulations of the Atlantic Ocean during the period 1950 to 1989, using a sigma coordinate, free surface numerical model, show long-term variabilities in the upper ocean subtropical gyre similar to those obtained from observations. The simulations show how westward propagating planetary waves, originated in the eastern North Atlantic, affect interdecadal variabilities of ocean properties such as the Bermuda sea level, the Gulf Stream position and strength, and subsurface temperature anomalies in the western North Atlantic. Special attention is given to the dramatic sea level drop at Bermuda in the early 1970s, which is accompanied by cooling of subsurface layers in the western North Atlantic and a northward shift and weakening of the Gulf Stream. Following these events, between 1970 and 1980, the cold temperature anomalies in the upper layers of the western North Atlantic slowly propagated eastward and downward; the strongest propagating signal in the model is found at 200-m depth, suggesting that advection of anomalies downstream by the Gulf Stream current and changes in winter mixing are involved. Significant correlations were found between the sea level anomalies at Bermuda and sea level anomalies in the eastern North Atlantic up to eight years earlier. Sensitivity experiments with different atmospheric forcing fields are used to study the ocean response to observed sea surface temperature and wind stress anomalies. It is shown that on decadal timescales, the ocean model responds in a linear fashion to the combined effect of SST and wind stress anomalies, a fact that might be exploited in future climate prediction studies

Can the Gulf Stream Induce Coherent Short-Term Fluctuations in Sea level along the US East Coast?: A Modeling Study

Much attention has been given in recent years to observations and models that show that variations in the transport of the Atlantic Meridional Overturning Circulation (AMOC) and in the Gulf Stream (GS) can contribute to interannual, decadal, and multi-decadal variations in coastal sea level (CSL) along the US East Coast. However, less is known about the impact of short-term (time scales of days to weeks) fluctuations in the GS and their impact on CSL anomalies. Some observations suggest that these anomalies can cause unpredictable minor tidal flooding in low-lying areas when the GS suddenly weakens. Can these short-term CSL variations be attributed to changes in the transport of the GS? An idealized numerical model of the GS has been set up to test this proposition. The regional model uses a 1/12° grid with a simplified coastline to eliminate impacts from estuaries and small-scale coastal features and thus isolate the GS impact. The GS in the model is driven by inflows/outflows, representing the Florida Current (FC), the Slope Current (SC), and the Sargasso Sea (SS) flows. Forcing the model with an oscillatory FC transport with a period of 2, 5, and 10 days produced coherent CSL variations from Florida to the Gulf of Maine with similar periods. However, when imposing variations in the transports of the SC or the SS, they induce CSL variations only north of Cape Hatteras. The suggested mechanism is that variations in GS transport produce variations in sea level gradient across the entire GS length and this large-scale signal is then transmitted into the shelf by the generation of coastal-trapped waves (CTW). In this idealized model, the CSL variations induced by variations of ∼10 Sv in the transport of the GS are found to resemble CSL variations induced by ∼5ms−1 zonal wind fluctuations, though the mechanisms of wind-driven and GS-driven sea level are quite different. Better understanding of the relation between variations in offshore currents and CSL will help to improve the prediction of both short-term water level anomalies that cause flooding, as well as spatial variations in long-term sea level variability and coastal sea level rise

Comments on Reconsidering the Relationship Between Gulf Stream Transport and Dynamic Sea Level at U.S. East Coast by Chi et al.

Numerous recent studies found significant correlations between weakening of the Gulf Stream (GS) and rising coastal sea level (CSL) along the U.S. East Coast. Based on monthly altimeter data and Florida Current transport, Chi et al. (2023; here, CH23) argued that geostrophic adjustment of the GS is unlikely to drive variations in CSL in the Mid-Atlantic Bight (MAB). It is argued here that this conclusion cannot be universally applicable to all cases, since the monthly data disregard correlations previously found for short time scales based on hourly and daily data; the impact of GS variability on time scales of decades and longer as well as potential time lags between the GS and CSL variability were also not considered by CH23. Examples are given here to demonstrate the important role of the GS in post hurricane coastal flooding

ODU-European Collaborations on Climate Change and Sea Level Rise Reserach

Less than five years ago, Old Dominion University started the Climate Change and Sea Level Rise Initiative (CCSLRI), which led to the recently established Mitigation and Adaptation Research Institute (MARI) and the Hampton Roads Sea Level Rise Preparedness & Resilience Intergovernmental Planning Pilot Project. This interdisciplinary area of research also has a long history in many European countries. Direct measurements of sea level started more than 200 years ago and flood mitigation measures have been in effect for a long time in London, the Netherlands and many other places. Today, reports on flooding in Norfolk, UK, by the BBC or reports on flooding in Norfolk, Va., USA, by the Washington Post, are eerily similar. Therefore, studies of sea level rise (SLR) and associated flooding must be a collaborative effort across oceanic boundaries