Numerical solutions to the parabolic wave equation are plagued by the curse of dimensionality coupled with the Nyquist criterion. As a remedy, a new range-dynamical low-rank split-step Fourier method is developed. The integration scheme scales sub-linearly with the number of classical degrees of freedom in the transverse directions. It is orders of magnitude faster than the classic full-rank split-step Fourier algorithm and saves copious amounts of storage space. This enables numerical solutions of the parabolic wave equation at higher frequencies and on larger domains, and simulations may be performed on laptops rather than high-performance computing clusters. Using a rank-adaptive scheme to optimize the low-rank equations further ensures the approximate solution is highly accurate and efficient. The methodology and algorithms are demonstrated on realistic high-resolution data-assimilative ocean fields in Massachusetts Bay for two three-dimensional acoustic configurations with different source locations and frequencies. The acoustic pressure, transmission loss, and phase solutions are analyzed in the two geometries with seamounts and canyons across and along Stellwagen Bank. The convergence with the rank of the subspace and the properties of the rank-adaptive scheme are demonstrated, and all results are successfully compared with those of the full-rank method when feasible.

The Bay of Bengal (BoB) exhibits a distinctive pattern of surface freshening primarily resulting from runoff originating from several major rivers and the monsoon precipitation. This freshening significantly modulates the spatial and temporal variations in the thermohaline structure, ultimately shaping the sound speed structure within this region. This study investigates the seasonal impact of river input on the sound speed structure of the BoB through two numerical simulations with and without river input using the Regional Ocean Modeling System (ROMS). The findings indicate that river inputs consistently reduce the surface sound speed across the domain throughout the year, with the most noticeable effect occurring in the northern part of BoB during the post-monsoon months of October and November. During this period, the surface variability is predominately driven by salinity variations induced by river inputs. In contrast, in the subsurface layers, the influence of reduced salinity becomes less pronounced with increasing depth, and the temperature modulations brought about by river inputs play a more important role. Freshening in the surface layers leads to the creation of a stratified barrier layer just below the mixed layer. Consequently, this results in the formation of warm temperature inversions in the subsurface layers, with cooling occurring beneath them. These phenomena contribute to variations in the sound speed, causing it to increase within the inversion layer and decrease below it. Notably, the sonic layer depth (SLD) is found to become shallower in the presence of river inputs during the post-monsoon and winter seasons in the northern BoB. The combination of enhanced vertical salinity gradients and subsurface temperature inversions significantly amplifies the vertical gradient of sound speed above the SLD. This, in turn, may lead to the development of more robust surface ducts and the expansion of shadow zones beneath the SLD.

The stochastic dynamically orthogonal (DO) narrow-angle parabolic equations (NAPEs) are exemplified and their properties and capabilities are described using three new 2D stochastic range-independent and range-dependent test cases with uncertain sound speed field, bathymetry, and source location. We validate results against ground-truth deterministic analytical solutions and direct Monte Carlo predictions of acoustic pressure and transmission loss fields. We verify the stochastic convergence and computational advantages of the DO-NAPEs and discuss the differences with normal mode approaches. Results show that a single DO-NAPE simulation can accurately predict stochastic range-dependent acoustic fields and their non-Gaussian probability distributions, with computational savings of several orders of magnitude when compared to direct Monte Carlo methods. With their coupling properties and their adaptation in range to the dominant uncertainties, the DO-NAPEs are shown to predict accurate statistics, from mean and variance to multiple modes and full probability distributions, and to provide excellent reconstructed realizations, from amplitudes and phases to other specific properties of complex realization fields.

Robust informative acoustic predictions require precise knowledge of ocean physics, bathymetry, seabed, and acoustic parameters. However, in realistic applications, this information is uncertain due to sparse and heterogeneous measurements and complex ocean physics. Efficient techniques are thus needed to quantify these uncertainties and predict the stochastic acoustic wave fields. In this work, we derive and implement new stochastic differential equations that predict the acoustic pressure fields and their probability distributions. We start from the stochastic acoustic parabolic equation (PE) and employ the instantaneously-optimal Dynamically Orthogonal (DO) equations theory. We derive stochastic DO-PEs that dynamically reduce and march the dominant multi-dimensional uncertainties respecting the nonlinear governing equations and non-Gaussian statistics. We develop the dynamical reduced-order DO-PEs theory for the Narrow-Angle PE (NAPE) and implement numerical schemes for discretizing and integrating the stochastic acoustic fields.