Temporal Evolution of Cooling by Natural Convection in an Enclosed Magma Chamber

This research numerically studies the transient cooling of partially liquid magma by natural convection in an enclosed magma chamber. The mathematical model is based on the conservation laws for momentum, energy and mass for a non-Newtonian and incompressible fluid that may be modeled by the power law and the Oberbeck–Boussinesq equations (for basaltic magma) and solved with the finite volume method (FVM). The results of the programmed algorithm are compared with those in the literature for a non-Newtonian fluid with high apparent viscosity (10–200 Pa s) and Prandtl (Pr = 4 × 104) and Rayleigh (Ra = 1 × 106) numbers yielding a low relative error of 0.11. The times for cooling the center of the chamber from 1498 to 1448 K are 40 ky (kilo years), 37 and 28 ky for rectangular, hybrid and quasi-elliptical shapes, respectively. Results show that for the cases studied, natural convection moved the magma but had no influence on the isotherms; therefore the main mechanism of cooling is conduction. When a basaltic magma intrudes a chamber with rhyolitic magma in our model, natural convection is not sufficient to effectively mix the two magmas to produce an intermediate SiO2 composition.

Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs

Consider a general linear Hamiltonian system ∂ t u = J L u \partial _{t}u=JLu in a Hilbert space X X . We assume that L : X → X ∗ \ L:X\rightarrow X^{\ast } induces a bounded and symmetric bi-linear form ⟨ L ⋅ , ⋅ ⟩ \left \langle L\cdot ,\cdot \right \rangle on X X , which has only finitely many negative dimensions n − ( L ) n^{-}(L) . There is no restriction on the anti-self-dual operator J : X ∗ ⊃ D ( J ) → X J:X^{\ast }\supset D(J)\rightarrow X . We first obtain a structural decomposition of X X into the direct sum of several closed subspaces so that L L is blockwise diagonalized and J L JL is of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy of e t J L e^{tJL} . In particular, e t J L e^{tJL} has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relate n − ( L ) n^{-}\left ( L\right ) and the dimensions of generalized eigenspaces of eigenvalues of J L \ JL , some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostly J J was assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schrödinger equations with nonzero conditions at infinity, where our general theory applies to yield stability or instability of some coherent states.

Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

<p style='text-indent:20px;'>In this article, we are concerned with statistical solutions for the nonautonomous coupled Schrödinger-Boussinesq equations on infinite lattices. Firstly, we verify the existence of a pullback-<inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor and establish the existence of a unique family of invariant Borel probability measures carried by the pullback-<inline-formula><tex-math id="M2">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor for this lattice system. Then, it will be shown that the family of invariant Borel probability measures is a statistical solution and satisfies a Liouville type theorem. Finally, we illustrate that the invariant property of the statistical solution is indeed a particular case of the Liouville type theorem.</p>

Exploiting self-organized criticality in strongly stratified turbulence

A multiscale reduced description of turbulent free shear flows in the presence of strong stabilizing density stratification is derived via asymptotic analysis of the Boussinesq equations in the simultaneous limits of small Froude and large Reynolds numbers. The analysis explicitly recognizes the occurrence of dynamics on disparate spatiotemporal scales, yielding simplified partial differential equations governing the coupled evolution of slow large-scale hydrostatic flows and fast small-scale isotropic instabilities and internal waves. The dynamics captured by the coupled reduced equations is illustrated in the context of two-dimensional strongly stratified Kolmogorov flow. A noteworthy feature of the reduced model is that the fluctuations are constrained to satisfy quasilinear (QL) dynamics about the comparably slowly varying large-scale fields. Crucially, this QL reduction is not invoked as an ad hoc closure approximation, but rather is derived in a physically relevant and mathematically consistent distinguished limit. Further analysis of the resulting slow–fast QL system shows how the amplitude of the fast stratified-shear instabilities is slaved to the slowly evolving mean fields to ensure the marginal stability of the latter. Physically, this marginal stability condition appears to be compatible with recent evidence of self-organized criticality in both observations and simulations of stratified turbulence. Algorithmically, the slaving of the fluctuation fields enables numerical simulations to be time-evolved strictly on the slow time scale of the hydrostatic flow. The reduced equations thus provide a solid mathematical foundation for future studies of three-dimensional strongly stratified turbulence in extreme parameter regimes of geophysical relevance and suggest avenues for new sub-grid-scale parametrizations.

Effects of aquifer geometry on seawater intrusion in annulus segment island aquifers

Seawater intrusion in island aquifers was considered analytically, specifically for annulus segment aquifers (ASAs), i.e., aquifers that (in plan) have the shape of an annulus segment. Based on the Ghijben–Herzberg and hillslope-storage Boussinesq equations, analytical solutions were derived for steady-state seawater intrusion in ASAs, with a focus on the freshwater–seawater interface and its corresponding watertable elevation. Predictions of the analytical solutions compared well with experimental data, and so they were employed to investigate the effects of aquifer geometry on seawater intrusion in island aquifers. Three different ASA geometries were compared: convergent (smaller side is facing the lagoon, larger side is the internal no-flow boundary and flow converges towards the lagoon), rectangular and divergent (smaller side is the internal no-flow boundary, larger side is facing the sea and flow diverges towards the sea). Depending on the aquifer geometry, seawater intrusion was found to vary greatly, such that the assumption of a rectangular aquifer to model an ASA can lead to poor estimates of seawater intrusion. Other factors being equal, compared with rectangular aquifers, seawater intrusion is more extensive, and watertable elevation is lower in divergent aquifers, with the opposite tendency in convergent aquifers. Sensitivity analysis further indicated that the effects of aquifer geometry on seawater intrusion and watertable elevation vary with aquifer width and distance from the circle center to the inner arc (the lagoon boundary for convergent aquifers or the internal no-flow boundary for divergent aquifers). A larger aquifer width and distance from the circle center to the inner arc weaken the effects of aquifer geometry, and hence differences in predictions for the three geometries become less pronounced.

Classification of All Single Traveling Wave Solutions of Fractional Coupled Boussinesq Equations via the Complete Discrimination System Method

In this paper, the complete discrimination system method is used to construct the exact traveling wave solutions for fractional coupled Boussinesq equations in the sense of conformable fractional derivatives. As a result, we get the exact traveling wave solutions of fractional coupled Boussinesq equations, which include rational function solutions, Jacobian elliptic function solutions, implicit solutions, hyperbolic function solutions, and trigonometric function solutions. Finally, the obtained solution is compared with the existing literature.