Gerardo Hernández-Dueñas

In this paper, we formulate a model for weakly compressible two-layer shallow water flows with friction in general channels. The formulated model is non-conservative, and in contrast to the incompressible limit, our system is strictly hyperbolic. The generalized Rankine–Hugoniot conditions are provided for the present system with non-conservative products to define weak solutions. We write the Riemann invariants along each characteristic field for channels with constant width in an appendix. A robust well-balanced path-conservative semi-discrete central-upwind scheme is proposed and implemented to validate exact solutions to the Riemann problem. We also present numerical tests in general channels to show the merits of the scheme.

Over the years, there have been discussions about the possibility of air pollution affecting the process of rain formation. In this study, we have developed a simplified model that represents the atmospheric dynamics and cloud microphysics to explore the impact of pollution on rain formation. We used an existing three-dimensional minimal model consisting of five equations, for which we added a simple bulk parametrization that represents the role of cloud condensation nuclei (CCN) in cloud formation processes. We conducted numerical tests using two CCN profiles, with either one or two accumulation layers and modified their abundance to explore the effects of different CCN concentrations and distributions. We conducted four numerical tests corresponding to the two aforementioned profiles with polluted and low-polluted scenarios. The numerical simulations suggested that a layer with high CCN concentration close to the surface tends to suppress precipitation, while the same concentration distributed over two layers tends to enhance the efficiency of rain formation. The simulations also showed that CCN particles far from the surface produced higher cloud tops and longer events, consistent with previous research. Although the model includes a stable representation of precipitating turbulent convection with bulk cloud microphysics, we expect its simplicity and conservation properties to allow for deeper theoretical analyses that can help us better understand the physical processes involved in the studied phenomenon. We hope this model will serve as a tool to explore different aerosol-related scenarios within the context of minimal models.

We consider a rotating non-hydrostatic flow with arbitrary stratification and argue that 1) the appropriate form of potential vorticity (PV) for this system is in terms of isopycnal deviation and 2) the decomposition into energetically orthogonal solutions is fundamentally a PV-inversion. The new closed-form expression for available potential vorticity (APV) is expressed in terms of isopycnal deviation, following the ideas in Wagner & Young (2015). This form of APV linearizes to quasigeostrophic PV (QGPV) after discarding the nonlinear stretching term and a height nonlinearity, the latter of which is not present in constant stratification. This formulation leads to positive definite definitions of potential enstrophy and total energy expressed in terms of isopycnal deviation, from which the quadratic versions emerge at lowest order. It is exactly these quantities diagonalized by the linear eigenmodes. Internal-gravity waves, geostrophic motions, inertial oscillations, and a mean density anomaly form the energetically and enstrophically orthogonal constituents of flow. The complete state of the fluid can be represented in terms of these physically realizeable modes and determined from the derived projection operators using the horizontal velocity and density anomaly. The projection of the fluid state onto the non-hydrostatic wave modes, reveals that one must first account for the PV portion of the flow before recovering the wave solutions. We apply the physical insights of the decomposition to a mesoscale eddy showing how strict adherence to adiabatic rearrangement places strong constraints on the vertical structure of such eddies, including a skew towards stronger cyclonic eddies in the upper water-column. Finally, the expression for APV is shown to reproduce the height nonlinearity of shallow-water PV, a well know feature that breaks the cyclone-anticyclone symmetry in QGPV.

A new two-dimensional model for blood flows in arteries with arbitrary cross sections is derived. The domain consists of a narrow, large vessel that extends along an axial direction, with cross sections described by radial and angular coordinates. The model consists of a system of balance laws for conservation of mass and balance of momentum in the axial and angular directions. The equations are derived by applying asymptotic analysis to the incompressible Navier-Stokes equations in a moving domain with an elastic membrane, and integrating in the radial direction in each cross section. The resulting model is a system of hyperbolic balance laws with source terms. The main properties of the system are discussed and a positivity-preserving well-balanced central-upwind scheme is presented. The merits of the scheme will be tested in a variety of scenarios. In particular, simulations using an idealized aorta model are shown. We analyze the time evolution of the blood flow under different initial conditions such as perturbations to steady states, which parametrizes a bulging in a vessel’s wall. We consider different situations given by distinct variations in the vessel’s elasticity.

We develop a transfer operator-based method for the detection of coherent structures and their associated lifespans. Characterising the lifespan of coherent structures allows us to identify dynamically meaningful time windows, which may be associated with transient coherent structures in the localised phase space, as well as with time intervals within which these structures experience fundamental changes, such as merging or separation events. The localised transfer operator approach we pursue allows one to explore the fundamental properties of a dynamical system without full knowledge of the dynamics. The algorithms we develop prove useful not only in the simple case of a periodically driven double well potential model, but also in more complex cases generated using the rotating Boussinesq equations.

The shallow water equations (SWE) model a variety of geophysical flows. Flows in channels with rectangular cross sections may be modeled with a simplified one-dimensional SWE with varying width. Among other model parameters, information about the bathymetry and friction coefficient is needed for the correct and precise prediction of the flow. Although synthetic values of the model parameters may suffice for testing numerical schemes, approximations of the bathymetry and other parameters may be required for specific applications. Estimations may be obtained by experimental methods, but some of those techniques may be expensive, time consuming, and not always available. In this work, we propose to solve the inverse problem to estimate the bathymetry and the Manning's friction coefficient from transient velocity data. This is done with the aid of a cost functional, which includes the SWE through Lagrange multipliers. We prove that the velocity data determine uniquely the derivative of the bathymetry in a linearized shallow water system. That is, the inverse problem is identifiable. The solution is obtained by solving the constrained optimization problem by a continuous descent method. The direct and the adjoint problems are both solved numerically using a second-order accurate Roe-type upwind scheme. Numerical tests are included to show the merits of the algorithm.

Submesoscale lateral transport of Lagrangian particles in pycnocline conditions is investigated by means of idealized numerical simulations with reduced-interaction models. Using a projection technique, the models are formulated in terms of wave-mode and vortical-mode nonlinear interactions, and they range in complexity from full Boussinesq to waves-only and vortical-modes-only (QG) models. We find that, on these scales, most of the dispersion is done by vortical motions, but waves cannot be discounted because they play an important, albeit indirect, role. In particular, we show that waves are instrumental in filling out the spectra of vortical-mode energy at smaller scales through nonresonant vortex–wave–wave triad interactions. We demonstrate that a richer spectrum of vortical modes in the presence of waves enhances the effective lateral diffusivity, relative to QG. Waves also transfer energy upscale to vertically sheared horizontal flows that are a key ingredient for internal-wave shear dispersion. In the waves-only model, the dispersion rate is an order of magnitude smaller and is attributed entirely to internal-wave shear dispersion.

We present a new high-resolution, non-oscillatory semi-discrete central-upwind scheme for one-dimensional two-layer shallow-water flows with friction and entrainment along channels with arbitrary cross sections and bottom topography. These flows are described by a conditionally hyperbolic balance law with non-conservative products. A detailed description of the properties of the model is provided, including entropy inequalities and asymptotic approximations of the eigenvalues of the corresponding coefficient matrix. The scheme extends existing central-upwind semi-discrete numerical methods for hyperbolic conservation and balance laws and it satisfies two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water depth for each layer, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. Along with the description of the scheme and proofs of these two properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

Summary: In this work, we consider a hyperbolic one‐dimensional (1D) model for blood flow through compliant axisymmetric tilted vessels. The pressure is a function of the cross‐sectional area and other model parameters. Important features of the model are inherited at the discrete level by the numerical scheme. For instance, the existence of steady states may provide important information about the flow properties at low computational cost. Here, we characterize a large class of smooth equilibrium solutions by means of quantities that remain invariant. At the discrete level, the well‐balanced property in the numerical scheme leads to accurate results when steady states are perturbed. On the other hand, the model is equipped with an entropy function and an entropy inequality that can help us choose the physically relevant weak solutions. A large class of semidiscrete entropy‐satisfying numerical schemes is described. In addition, preservation of positivity for the cross‐sectional area is achieved. Numerical results show the scheme is robust, stable, and accurate. The ultimate goal of this article is the numerical application to cases that are more relevant from the medical viewpoint. In particular, a numerical simulation of cardiac cycles with appropriate parameters shows that increasing the rigidity of the artery walls delays the formation of shock waves. Gravity effects are also measured in tilted vessels, and a simulation using an idealized aorta model was conducted.