Generalized Quantum Integro-Differential Fractional Operator with Application of 2D-Shallow Water Equation in a Complex Domain

In this paper, we aim to generalize a fractional integro-differential operator in the open unit disk utilizing Jackson calculus (quantum calculus or q-calculus). Next, by consuming the generalized operator to define a formula of normalized analytic functions, we present a set of integral inequalities using the concepts of subordination and superordination. In addition, as an application, we determine the maximum and minimum solutions of the extended fractional 2D-shallow water equation in a complex domain.

A unified moisture mode theory for the Madden Julian Oscillation and the Boreal Summer Intraseasonal Oscillation

The Madden Julian Oscillation (MJO) and the Boreal Summer Intraseasonal Oscillation (BSISO) are fundamental modes of variability in the tropical atmosphere on the intraseasonal time scale. A linear model, using a moist shallow water equation set on an equatorial beta plane, is developed to provide a unified treatment of the two modes and to understand their growth and propagation over the Indian Ocean. Moisture is assumed to increase linearly with longitude and to decrease quadratically with latitude. Solutions are obtained through linear stability analysis, considering the gravest (n = 1) meridional mode with nonzero meridional velocity. Anomalies in zonal moisture advection and surface fluxes are both proportional to those in zonal wind, but of opposite sign. With observation-based estimates for both effects, the zonal advection dominates, and drives the planetary-scale instability. With a sufficiently small meridional moisture gradient, the horizontal structure exhibits oscillations with latitude and a northwest-southeast horizontal tilt in the northern hemisphere, qualitatively resembling the observed BSISO. As the meridional moisture gradient increases, the horizontal tilt decreases and the spatial pattern transforms toward the “swallowtail” structure associated with the MJO, with cyclonic gyres in both hemispheres straddling the equatorial precipitation maximum. These results suggest that the magnitude of the meridional moisture gradient shapes the horizontal structures, leading to the transformation from the BSISO-like tilted horizontal structure to the MJO-like neutral wave structure as the meridional moisture gradient changes with the seasons. The existence and behavior of these intraseasonal modes can be understood as a consequence of phase speed matching between the equatorial mode with zero meridional velocity (analogous to the dry Kelvin wave) and a local off-equatorial component that is characterized by considering an otherwise similar system on an f-plane.

A Weighted-Least-Squares Meshless Model for Non-Hydrostatic Shallow Water Waves

In this paper, an explicit time marching procedure for solving the non-hydrostatic shallow water equation (SWE) problems is developed. The procedure includes a process of prediction and several iterations of correction. In these processes, it is essential to accurately calculate the spatial derives of the physical quantities such as the temporal water depth, the average velocities in the horizontal and vertical directions, and the dynamic pressure at the bottom. The weighted-least-squares (WLS) meshless method is employed to calculate these spatial derivatives. Though the non-hydrostatic shallow water equations are two dimensional, on the focus of presenting this new time marching approach, we just use one dimensional benchmark problems to validate and demonstrate the stability and accuracy of the present model. Good agreements are found in the comparing the present numerical results with analytic solutions, experiment data, or other numerical results.

Modeling The Hydrodynamic Characteristics of Tidal And Monsoonal Currents In Pondok Dayung Port of Tanjung Priok Harbor, Jakarta

The Pondok Dayung port forms a significant segment of the Tanjung Priok harbor in the Jakarta coastal bay. Studies on the hydrodynamic characteristics of tidal and monsoonal currents appear very important to ship movement and laid/dock operations in port basins/jetties. These flow conditions have been simulated using a two-dimensional shallow water equation, while the tidal and monsoonal wind were coupled to model the ocean current. In general, the simulation results of the ocean current characteristics were dominated by tidal effects, as well as the interactions with the coastlines, jetties, and breakwaters. Also, the geometric replica has been validated satisfactorily, using time series sea elevation from the tidal station in the research area managed by the National Geospatial Agency (BIG). Strong RMSE and linear correlation values ranging from 0.0405-0.0458 m and 0.9648-0.9843 were obtained, respectively. During the flood tides, the ocean current is directed towards the basin area, while an outward flow is observed under ebb conditions. Furthermore, the maximum tidal current speed of ±0.26 m/s was recorded at the port waterways. A similar outcome was also reported during the west and east monsoon, in addition to a minimum ocean current speed of approximately 0.00 m/s. These conditions implied that the Pondok Dayung port and its breakwater system served as protective structures to the surrounding vessels and the harsh ocean current impacts.

Automated Detection of Instability-Inducing Channel Geometry Transitions in Saint-Venant Simulation of Large-Scale River Networks

A new sweep-search algorithm (SSA) is developed and tested to identify the channel geometry transitions responsible for numerical convergence failure in a Saint-Venant equation (SVE) simulation of a large-scale open-channel network. Numerical instabilities are known to occur at “sharp” transitions in discrete geometry, but the identification of problem locations has been a matter of modeler’s art and a roadblock to implementing large-scale SVE simulations. The new method implements techniques from graph theory applied to a steady-state 1D shallow-water equation solver to recursively examine the numerical stability of each flowpath through the channel network. The SSA is validated with a short river reach and tested by the simulation of ten complete river systems of the Texas–Gulf Coast region by using the extreme hydrological conditions recorded during hurricane Harvey. The SSA successfully identified the problematic channel sections in all tested river systems. Subsequent modification of the problem sections allowed stable solution by an unsteady SVE numerical solver. The new SSA approach permits automated and consistent identification of problem channel geometry in large open-channel network data sets, which is necessary to effectively apply the fully dynamic Saint-Venant equations to large-scale river networks or for city-wide stormwater networks.

Simulasi Numerik Persamaan Gelombang Air Dangkal untuk Kasus Bendungan Bobol

Technological failure and natural disasters that caused the dam-break resulted in huge losses, both material loss and loss of life. The mathematical model for the dam-break can use the shallow water equation. In this paper, modeling the dam-break in two dimensions is solved by using the finite volume method with a stagerred-grid scheme. The staggered-grid scheme produces more accurate and robust when compared to the Lax-Friedrics scheme. The stability of the water waves on the part of the damaged dam wall is also well preserved using a staggered-grid scheme. Modeling a dam-break with real bathymetric data will be a challenge for further research, because it involves a more complex geometry.