An official website of the United States government
Flow configurations that maximize the instantaneous rate of conversion from potential to kinetic energy are sought using a combination of analytical and numerical methods. A hydrostatic model is briefly investigated, but the presence of unrealistic ageostrophic flow configurations renders the results unrealistic. In the quasigeostrophic (QG) model, flow configurations that locally optimize the conversion rate are found, but it remains unclear if these flow configurations produce the global maximum conversion rate. The difficulty is associated with the fact that in the QG model, the vertical velocity is a quadratic function of the QG streamfunction, which renders the conversion rate a cubic function of the QG streamfunction. For these locally maximal conversion rates, the rate of conversion depends on the horizontal length scale of the flow: for scales larger than the deformation radius, the maximal rates are small and decrease as the horizontal scale increases; for scales smaller than the deformation radius, the maximal conversion rate rises until it becomes comparable to the maximal rate at which potential energy can be extracted from the mean flow.
more »
« less
On energy exchanges between eddies and the mean flow in quasigeostrophic turbulence
We study the term in the eddy energy budget of continuously stratified quasigeostrophic turbulence that is responsible for energy extraction by eddies from the background mean flow. This term is a quadratic form, and we derive Euler–Lagrange equations describing its eigenfunctions and eigenvalues, the former being orthogonal in the energy inner product and the latter being real. The eigenvalues correspond to the instantaneous energy growth rate of the associated eigenfunction. We find analytical solutions in the Eady problem. We formulate a spectral method for computing eigenfunctions and eigenvalues, and compute solutions in Phillips-type and Charney-type problems. In all problems, instantaneous growth is possible at all horizontal scales in both inviscid problems and in problems with linear Ekman friction. We conjecture that transient growth at small scales is matched by linear transfer to decaying modes with the same horizontal structure, and we provide simulations supporting the plausibility of this hypothesis. In Charney-type problems, where the linear problem has exponentially growing modes at small scales, we expect net energy extraction from the mean flow to be unavoidable, with an associated nonlinear transfer of energy to dissipation.
more »
« less
- Award ID(s):
- 1736708
- PAR ID:
- 10151218
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 885
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
A discussion of three-wave interaction systems with rapidly decaying data is provided. Included are the classical and two nonlocal three-wave interaction systems. These three-wave equations are formulated from underlying compatible linear systems and are connected to a third order linear scattering problem. The inverse scattering transform (IST) is carried out in detail for all these three-wave interaction equations. This entails obtaining and analyzing the direct scattering problem, discrete eigenvalues, symmetries, the inverse scattering problem via Riemann--Hilbert methods, minimal scattering data, and time dependence. In addition, soliton solutions illustrating energy sharing mechanisms are also discussed. A crucial step in the analysis is the use of adjoint eigenfunctions which connects the third order scattering problem to key eigenfunctions that are analytic in the upper/lower half planes. The general compatible nonlinear wave system and its classical and nonlocal three-wave reductions are asymptotic limits of physically significant nonlinear equations, including water/gravity waves with surface tension.more » « less
-
null (Ed.)We discuss computational and qualitative aspects of the fractional Plateau and the prescribed fractional mean curvature problems on bounded domains subject to exterior data being a subgraph. We recast these problems in terms of energy minimization, and we discretize the latter with piecewise linear finite elements. For the computation of the discrete solutions, we propose and study a gradient flow and a Newton scheme, and we quantify the effect of Dirichlet data truncation. We also present a wide variety of numerical experiments that illustrate qualitative and quantitative features of fractional minimal graphs and the associated discrete problems.more » « less
-
Sixth-order boundary value problems (BVPs) arise in thin-film flows with a surface that has elastic bending resistance. We consider the case in which the elastic interface is clamped at the lateral walls of a closed trough and thus encloses a finite amount of fluid. For a slender film undergoing infinitesimal deformations, the displacement of the elastic surface from its initial equilibrium position obeys a sixth-order (in space) initial boundary value problem (IBVP). To solve this IBVP, we construct a set of odd and even eigenfunctions that intrinsically satisfy the boundary conditions (BCs) of the original IBVP. These eigenfunctions are the solutions of a non-self-adjoint sixth-order eigenvalue problem (EVP). To use the eigenfunctions for series expansions, we also construct and solve the adjoint EVP, leading to another set of even and odd eigenfunctions, which are orthogonal to the original set (biorthogonal). The eigenvalues of the adjoint EVP are the same as those of the original EVP, and we find accurate asymptotic formulas for them. Next, employing the biorthogonal sets of eigenfunctions, a Petrov–Galerkin spectral method for sixth-order problems is proposed, which can also handle lower-order terms in the IBVP. The proposed method is tested on two model sixth-order BVPs, which admit exact solutions. We explicitly derive all the necessary formulas for expanding the quantities that appear in the model problems into the set(s) of eigenfunctions. For both model problems, we find that the approximate Petrov–Galerkin spectral solution is in excellent agreement with the exact solution. The convergence rate of the spectral series is rapid, exceeding the expected sixth-order algebraic rate.more » « less
-
The flow of a thin viscous liquid layer under an elastic film arises in natural processes, such as magmatic intrusions between rock strata, and industrial applications, such as coating surfaces with cured polymeric films. We study the linear dynamics of small perturbations to the equilibrium state of the film in a closed trough (finite film). Specifically, we are interested in the spreading (early-time) and leveling (late-time) dynamics as the film adjusts to equilibrium, starting from different initial perturbations. We consider both smooth and non-smooth spatially symmetric and localized initial conditions (perturbations). We find the exact series solutions for the film height, using the sixth-order complete orthonormal eigenfunctions associated with the posed initial-boundary-value problem derived in our previous work [Papanicolaou N C and Christov I C 2023J. Phys.: Conf. Ser.2675012016]. We show that the evolution of the perturbations begins with the spreading of the localized perturbations, followed by their mutual interaction as they spread, and finally, interactions with the confining lateral boundaries of the domain as the perturbations level. In particular, we highlight how the leading eigenvalues of the problem determine the scalings of certain figures of merit with time.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2019.969
