An analytical derivation of ice-shelf basal melt based on the dynamics of meltwater plumes

The interaction between ice shelves and the ocean is an important process for the development of marine ice sheets. However, it is difficult to model in full detail due to the high computational cost of coupled ice-ocean simulations, so that simplified basal-melt parameterizations are required. In this work, a new analytical expression for basal melt is derived from the theory of buoyant meltwater plumes moving upward under the ice shelf and driving the overturning circulation within the ice-shelf cavity. The governing equations are nondimensionalized in the case of an ice shelf with constant basal slope and uniform ambient ocean conditions. An asymptotic analysis of these equations in terms of small slopes and small thermal driving, assumed typical for Antarctic ice shelves, leads to an equation that can be solved analytically for the dimensionless melt rate. This analytical expression describes a universal melt-rate curve onto which the scaled results of the original plume model collapse. Its key features are a positive melt peak close to the grounding line and a transition to refreezing further away. Comparing the analytical expression with numerical solutions of the plume model generally shows a close agreement between the two, even for more general cases than the idealized geometry considered in the derivation. The results show how the melt rates adapt naturally to changes in the geometry and ambient ocean temperature. The new expression can readily be used for improving ice-sheet models that currently still lack a sufficiently realistic description of basal melt

Nonlinear asymptotic impedance model for a Helmholtz resonator liner

a b s t r a c t The usual nonlinear corrections for a Helmholtz resonator type impedance do not seem to be based on a systematic asymptotic solution of the pertaining equations. We aim to present a systematic derivation of a solution of the nonlinear Helmholtz resonator equation, in order to obtain analytically expressions for impedances close to resonance, while including nonlinear effects. The amplitude regime considered is such that when we stay away from the resonance condition, the nonlinear terms are relatively small and the solution obtained is of the linear equation (formed after neglecting the nonlinear terms). Close to the resonance frequency, the nonlinear terms can no longer be neglected and algebraic equations are obtained that describe the corresponding nonlinear impedance. Sample results are presented including a few comparisons with measurements available in the literature. The validity of the model is understood in the near resonance and nonresonance regimes

The Webster Equation Revisited

The problem of low-frequency sound propagation in slowly varying ducts is systematically analysed as a perturbation problem of slow variation. The Webster equation and some variants are derived, and the entrance/exit plane boundary layer is given. It is shown why a varying lined duct in general does not have a solution

Correction to:Numerical and asymptotic solutions of the pridmore-brown equation (AIAA Journal, 10.2514/1.J059140)

Correction Notice 1. This correction pertains to the word “uniform” in the paragraph just before section IIB, 12 lines below Equation (4), in the original article when it was first published online [https://doi.org/10.2514/1.J059140]. In detail: “This is almost, but not exactly, the case with uniform mean flow,::: ” should be “This is almost, but not exactly, the case with nonuniform mean flow,::: ”. Correction Notice 2. This correction pertains to the conclusion, following Equation (56), that “p0 constant” is the only solution of Equation (55), in the original article when it was first published online [https://doi.org/10.2514/1.J059140]. In detail: The solution of interest of Equation (55) is p0 constant. Its validity can simply be verified by substitution. However, the given proof that this solution is unique, is incomplete. Although we never use this uniqueness, and all that follows remains the same, we include this correction to avoid any confusion. The argument following Equation (56), that (formula presented) implies j∇p0 j 0 and thus p0 constant, is complete for any case where Re 0 is real, or u0 is uniform). For other cases, p0 constant is still the only eligible solution of Equation (55), but as yet we have not been able to prove its uniqueness

Numerical and asymptotic solutions of the pridmore-brown equation

A study is made of acoustic duct modes in two-dimensional and axisymmetric three-dimensional lined ducts with an isentropic inviscid transversely nonuniform mean flow and sound speed. These modes are described by a one-dimensional eigenvalue problem consisting of a Pridmore-Brown equation complemented by hard-wall or impedance-wall boundary conditions. A numerical solution, based on a Galerkin projection and an efficient method for the resulting nonlinear eigenvalue problem, is compared with analytical approximations for low and high frequencies. A collection of results is presented and discussed. Modal wave numbers are traced in the complex plane for varying impedance, showing the usual regular modes and surface waves. A study of a vanishing boundary layer (the Ingard limit) showed that, in contrast to the smoothly converging acoustic modes and downstream running acoustic surface wave, the convergence of the other surface waves is numerically more difficult. Effects of (transverse) turning points and exponential decay are discussed. Especially the occurrence of modes insensitive to the wall impedance is pointed out. Cut-on wave numbers of hard-wall modes are presented as a function of frequency. A strongly nonuniform mean flow gives rise to considerable differences between the modal behavior for low and for high frequencies