Lagrangian Mean Research Articles

Generalized Lagrangian mean curvature flows: The cotangent bundle case

Abstract In [18], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel ( n , 0 ) {(n,0)} -form, just like the Calabi–Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.

Lagrangian approach to interfacial water waves with free surface

This is a theoretical study on the interfacial water waves with a free surface in a two-layer system, where the lower fluid is taken to be heavier than the upper one. Lagrangian matching conditions are introduced for the physical fields separated by the interface. And a perturbation analysis is carried out to the third order to find the particle trajectory in the Lagrangian description. Observing the derived solution, a symbolic computation is introduced for obtaining the fifth order solution. The Lagrangian drifts, wave frequency, and set-up are also given as part of the solutions. Discontinuities across the interface are found for all of these physical quantities. A generalized set-up effect is found that the Lagrangian mean levels come near to both of the free surface and internal interface. Through some numerical calculations, it is shown that the larger density difference or relative wave height results in the larger drift velocity. The direction of particle motion in the upper layer is anti-clockwise in contrary to that in the lower layer. Better convergence for the Lagrangian solution than the Eulerian one is numerically demonstrated for the barotropic mode that the wave motion is dominated on the free surface.

On short time existence of Lagrangian mean curvature flow

We consider a short time existence problem motivated by a conjecture of Joyce (Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow. arXiv:1401.4949, 2014). Specifically we prove that given any compact Lagrangian Lsubset mathbb {C}^n with a finite number of singularities, each asymptotic to a pair of non-area-minimising, transversally intersecting Lagrangian planes, there is a smooth Lagrangian mean curvature flow existing for some positive time, that attains L as t searrow 0 as varifolds, and smoothly locally away from the singularities.

The space of compact self-shrinking solutions to the Lagrangian mean curvature flow in ℂ2{\mathbb{C}^{2}}

Abstract Let F n F_{n} : (Σ, h n h_{n} ) → \to ℂ 2 \mathbb{C}^{2} be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics { \{ h n h_{n} } \} converges smoothly to a Riemannian metric h. We show that a subsequence of { \{ F n F_{n} } \} converges smoothly to a branched conformally immersed Lagrangian self-shrinker F ∞ F_{\infty} : (Σ, h) → \to ℂ 2 \mathbb{C}^{2} . When the area bound is less than 16π, the limit F ∞ {F_{\infty}} is an embedded torus. When the genus of Σ is one, we can drop the assumption on convergence h n h_{n} → \to h. When the genus of Σ is zero, we show that there is no branched immersion of Σ as a Lagrangian self-shrinker, generalizing the rigidity result of [21] in dimension two by allowing branch points.

Investigation of micromixing by acoustically oscillated sharp-edges.

Recently, acoustically oscillated sharp-edges have been utilized to achieve rapid and homogeneous mixing in microchannels. Here, we present a numerical model to investigate acoustic mixing inside a sharp-edge-based micromixer in the presence of a background flow. We extend our previously reported numerical model to include the mixing phenomena by using perturbation analysis and the Generalized Lagrangian Mean (GLM) theory in conjunction with the convection-diffusion equation. We divide the flow variables into zeroth-order, first-order, and second-order variables. This results in three sets of equations representing the background flow, acoustic response, and the time-averaged streaming flow, respectively. These equations are then solved successively to obtain the mean Lagrangian velocity which is combined with the convection-diffusion equation to predict the concentration profile. We validate our numerical model via a comparison of the numerical results with the experimentally obtained values of the mixing index for different flow rates. Further, we employ our model to study the effect of the applied input power and the background flow on the mixing performance of the sharp-edge-based micromixer. We also suggest potential design changes to the previously reported sharp-edge-based micromixer to improve its performance. Finally, we investigate the generation of a tunable concentration gradient by a linear arrangement of the sharp-edge structures inside the microchannel.

Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in Cm

We prove two main results: (a) Suppose $L$ is a closed, embedded, exact special Lagrangian $m$-fold in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $\Pi_1\cup\Pi_2$ of two transverse special Lagrangian planes $\Pi_1,\Pi_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit 'Lawlor neck' family of examples found by Lawlor (Invent. math. 95, 1989). (b) Suppose $L$ is a closed, embedded, exact Lagrangian mean curvature flow expander in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $\Pi_1\cup\Pi_2$ of two transverse Lagrangian planes $\Pi_1,\Pi_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit family of examples found by Joyce, Lee and Tsui, arXiv:0801.3721. If instead $L$ is immersed rather than embedded, the only extra possibility in (a),(b) is $L=\Pi_1\cup\Pi_2$. Our methods, which are new and can probably be used to prove other similar uniqueness theorems, involve $J$-holomorphic curves, Lagrangian Floer cohomology, and Fukaya categories from symplectic topology. When $m=2$, (a) is easy to prove using hyperkahler geometry, and (b) is proved by Lotay and Neves, arXiv:1208.2729.

Lagrangian flows within reflecting internal waves at a horizontal free-slip surface

In this paper sequel to Zhou and Diamessis [“Reflection of an internal gravity wave beam off a horizontal free-slip surface,” Phys. Fluids 25, 036601 (2013)], we consider Lagrangian flows within nonlinear internal waves (IWs) reflecting off a horizontal free-slip rigid lid, the latter being a model of the ocean surface. The problem is approached both analytically using small-amplitude approximations and numerically by tracking Lagrangian fluid particles in direct numerical simulation (DNS) datasets of the Eulerian flow. Inviscid small-amplitude analyses for both plane IWs and IW beams (IWBs) show that Eulerian mean flow due to wave-wave interaction and wave-induced Stokes drift cancels each other out completely at the second order in wave steepness A, i.e., O(A2), implying zero Lagrangian mean flow up to that order. However, high-accuracy particle tracking in finite-Reynolds-number fully nonlinear DNS datasets from the work of Zhou and Diamessis suggests that the Euler-Stokes cancelation on O(A2) is not c...

Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow

Let (M,J,g,\Omega) be a Calabi–Yau m -fold, and consider compact, graded Lagrangians L in M . Thomas and Yau conjectured that there should be a notion of 'stability' for such L , and that if L is stable then Lagrangian mean curvature flow \{L^t:t\in[0,\infty)\} with L^0=L should exist for all time, and L^\infty=\lim_{t\to \infty}L^t should be the unique special Lagrangian in the Hamiltonian isotopy class of L . This paper is an attempt to update the Thomas–Yau conjectures, and discuss related issues. It is a folklore conjecture, extending [81], that there exists a Bridgeland stability condition (Z, \mathcal P) on the derived Fukaya category D^b \mathcal F (M) , such that an isomorphism class in D^b \mathcal F (M) is (Z, \mathcal P) -semistable if (and possibly only if) it contains a special Lagrangian, which must then be unique. In brief, we conjecture that if (L,E,b) is an object in an enlarged version of D^b \mathcal F (M) , then there is a unique family \{(L^t,E^t,b^t):t\in[0,\infty)\} such that (L^0,E^0,b^0)=(L,E,b) , and (L^t,E^t,b^t)\cong(L,E,b) in D^b \mathcal F (M) for all t , and \{L^t:t\in[0,\infty)\} satisfies Lagrangian MCF with surgeries at singular times T_1,T_2,\ldots, and in graded Lagrangian integral currents we have \lim_{t\to\infty}L^t=L_1+\cdots+L_n , where L_j is a special Lagrangian integral current of phase e^{i\pi\phi_j} for \phi_1>\cdots>\phi_n , and (L_1,\phi_1),\ldots,(L_n,\phi_n) correspond to the decomposition of (L,E,b) into (Z,P) -semistable objects. We also give detailed conjectures on the nature of the singularities of Lagrangian MCF that occur at the finite singular times T_1,T_2,\ldots.

On the second boundary value problem for Lagrangian mean curvature flow

We consider a fully nonlinear parabolic equation with nonlinear Neumann type boundary condition, and show the long time existence and convergence of the flow. Finally we apply this study to the boundary value problem for minimal Lagrangian graphs.

The slowly evolving background state of the atmosphere

The theory of wave–mean flow interaction requires a partition of the atmospheric flow into a notional background state and perturbations to it. Here, a background state, known as the Modified Lagrangian Mean (MLM), is defined as the zonally symmetric state obtained by requiring that every potential vorticity (PV) contour lying within an isentropic layer encloses the same mass and circulation as in the full flow. For adiabatic and frictionless flow, these two integral properties are time‐invariant and the MLM state is a steady solution of the primitive equations. The time dependence in the adiabatic flow is put into the perturbations, which can be described by a wave‐activity conservation law that is exact even at large amplitude. Furthermore, the effects of non‐conservative processes on wave activity can be calculated from the conservation law.A new method to calculate the MLM state is introduced, where the position of the lower boundary is obtained as part of the solution. The results are illustrated using Northern Hemisphere ERA‐Interim data. The MLM state evolves slowly, implying that the net non‐conservative effects are weak. Although ‘adiabatic eddy fluxes’ cannot affect the MLM state, the effects of Rossby‐wave breaking, PV filamentation and subsequent dissipation result in sharpening of the polar vortex edge and meridional shifts in the MLM zonal flow, both at tropopause level and on the winter stratospheric vortex. The rate of downward migration of wave activity during stratospheric sudden warmings is shown to be given by the vertical scale associated with polar vortex tilt divided by the time‐scale for wave dissipation estimated from the wave‐activity conservation law. Aspects of troposphere–stratosphere interaction are discussed.The new framework is suitable to examine the climate and its interactions with disturbances, such as midlatitude storm tracks, and makes a clean partition between adiabatic and non‐conservative processes.

Lagrangian solution for irrotational progressive water waves propagating on a uniform current: Part 1. Fifth-order analysis

A three-dimensional (3D) Lagrangian solution up to the fifth-order is found for the Boundary Value Problem (BVP) of irrotational, progressive water waves propagating in the presence of uniform current in water of constant depth. The BVP is formulated in the Lagrangian framework which uses Lagrangian label (a,b,c) and time t as the independent variables. The (a,b,c) of a particle in wave motion are the coordinates of the particle position in the still water. In the solution, wave–current interaction is embedded in the Lagrangian velocity potential in such way that pressure is not affected by current in the wave–current field. In this study, we present motion properties of particles such as the particle motion period, drift velocity, the Lagrangian mean level, and the 3D particle trajectory and streakline, aspects that are hardly described by the Eulerian solution. It is found that the particle motion period is always larger than the progressive wave period. As the wave–current becomes steady motion, the stable wavy track of the particle motion may exist in the wave–current field. When currents vanish, the present solution deduces to the same solution of the progressive wave propagation by Chen et al. (2010).

Mass Transport in the Stokes Edge Wave for Constant Arbitrary Bottom Slope in a Rotating Ocean

Abstract The Lagrangian mass transport in the Stokes surface edge wave is obtained from the vertically integrated equations of momentum and mass in a viscous rotating ocean, correct to the second order in wave steepness. The analysis is valid for bottom slope angles β in the interval 0 < β ≤ π/2. Vertically averaged drift currents are obtained by dividing the fluxes by the local depth. The Lagrangian mean current is composed of a Stokes drift (inherent in the waves) plus a mean Eulerian drift current. The latter arises as a balance between the radiation stresses, the Coriolis force, and bottom friction. Analytical solutions for the mean Eulerian current are obtained in the form of exponential integrals. The relative importance of the Stokes drift to the Eulerian current in their contribution to the Lagrangian drift velocity is investigated in detail. For the given wavelength, the Eulerian current dominates for medium and large values of β, while for moderate and small β, the Stokes drift yields the main contribution to the Lagrangian drift. Because most natural beaches are characterized by moderate or small slopes, one may only calculate the Stokes drift in order to assess the mean drift of pollution and suspended material in the Stokes edge wave. The main future application of the results for large β appears to be for comparison with laboratory experiments in rotating tanks.

On Langmuir circulation in shallow waters

AbstractThe instability of shallow-water waves on a moderate shear to Langmuir circulation is considered. In such instances, specifically at the shallow end of the inner coastal region, the shear can significantly affect the drift giving rise to profiles markedly different from the simple Stokes drift. Since drift and shear are instrumental in the instability to Langmuir circulation, of key interest is how that variation in turn affects onset to Langmuir circulation. Also of interest is the effect on onset of various boundary conditions. To that end the initial value problem describing the wave–mean flow interaction which accounts for the multiple time scales of the surface waves, evolving shear and evolving Langmuir circulation is crafted from scratch, and includes the wave-induced drift and a consistent set of free-surface boundary conditions. The problem necessitates that Navier–Stokes be employed side by side with a set of mean-field equations. Specifically, the former is used to evaluate events with the shortest time scale, that is the wave field, while the mean field set is averaged over that time scale. This averaged set, the CLg equations, follow from the generalized Lagrangian mean equations and for the case at hand take the same form as the well-known CL equations, albeit with different time and velocity scales. Results based upon the Stokes drift are used as a reference to which those based upon drift profiles corrected for shear are compared, noting that the latter are asymptotic to the former as the waves transition from shallow to deep. Two typical temporal flow fields are considered: shear-driven flow and pressure-driven flow. Relative to the reference case, shear-driven flow is found to be destabilizing while pressure driven are stabilizing to Langmuir circulation. In pressure-driven flows it is further found that multiple layers, as opposed to a single layer, of Langmuir circulation can form, with the most intense circulations at the ocean floor. Moreover, the layers can extend into a region of flow beyond that in which the instability applies, suggesting that Langmuir circulation excited by the instability can in turn drive, as a dynamic consequence, contiguous albeit less intense Langmuir circulation. Pressure-driven flows also admit two preferred spacings, one closely in accord with observation for small-aspect-ratio Langmuir circulation, the other well in excess of observed large-aspect-ratio Langmuir circulation.

Relating Lagrangian and Eulerian horizontal eddy statistics in the surfzone

AbstractConcurrent Lagrangian and Eulerian observations of rotational, low‐frequency (10−4 to 10−2 Hz) surfzone eddies are compared. Surface drifters were tracked for a few hours on each of 11 days at two alongshore uniform beaches. A cross‐shore array of near‐bottom current meters extended from near the shoreline to seaward of the surfzone (typically 100 m wide in these moderate wave conditions). Lagrangian and Eulerian mean alongshore velocities V are similar, with a midsurfzone maximum. Cross‐shore dependent Lagrangian (σL) and Eulerian (σE) rotational eddy velocities, estimated from low‐pass filtered drifter and current meter velocities, respectively, also generally agree. Cross‐shore rotational velocities have a midsurfzone maximum whereas alongshore rotational velocities are distributed more broadly. Daily estimates of the Lagrangian time scale, the time for drifter velocities to decorrelate, vary between 40 and 300 s, with alongshore time scales greater than cross‐shore time scales. The ratio of Lagrangian to apparent Eulerian current meter decorrelation times TL/TA varies considerably, between about 0.5 and 3. Consistent with theory, some of the TL/TA variation is ascribable to alongshore advection and TL/TA is proportional to V/σ, which ranges between about 0.6 and 2.5. Estimates of TL/TA vary between days with similar V/σ suggesting that surfzone Lagrangian particle dynamics vary between days, spanning the range from “fixed‐float” to “frozen‐field” [Lumpkin et al., 2002], although conclusions are limited by the statistical sampling errors in both TL/TA and V/σ.

Comment on “Wave‐turbulence interaction and its induced mixing in the upper ocean” by Huang and Qiao

Comment on “Wave‐turbulence interaction and its induced mixing in the upper ocean” by Huang and Qiao

A New Expression for the Form Stress Term in the Vertically Lagrangian Mean Framework for the Effect of Surface Waves on the Upper-Ocean Circulation

AbstractThere is an ongoing discussion in the community concerning the wave-averaged momentum equations in the hybrid vertically Lagrangian and horizontally Eulerian (VL) framework and, in particular, the form stress term (representing the residual effect of pressure perturbations) that is thought to restrict the handling of higher-order waves in terms of a perturbation expansion. The present study shows that the traditional pressure-based form stress term can be transformed into a set of terms that do not contain any pressure quantities but do contain the time derivative of a wave-induced velocity. This wave-induced velocity is referred to as the pseudomomentum in the VL framework, as it is analogous to the generalized pseudomomentum in Andrews and McIntyre. This enables the second expression for the wave-averaged momentum equations in the VL framework (this time for the development of the total transport velocity minus the VL pseudomomentum) to be derived together with the vortex force. The velocity-based expression of the form stress term also contains the residual effect of the turbulent viscosity, which is useful for understanding the dissipation of wave energy leading to a transfer of momentum from waves to circulation. It is found that the concept of the virtual wave stress of Longuet-Higgins is applicable to quite general situations: it does not matter whether there is wind forcing or not, the waves can have slow variations, and the viscosity coefficient can vary in the vertical. These results provide a basis for revisiting the surface boundary condition used in numerical circulation models.

Complete Self-Shrinking Solutions for Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space

Letf(x)be a smooth strictly convex solution ofdet(∂2f/∂xi∂xj)=exp(1/2)∑i=1nxi(∂f/∂xi)-fdefined on a domainΩ⊂Rn; then the graphM∇fof∇fis a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean spaceRn2nwith the indefinite metric∑dxidyi. In this paper, we prove a Bernstein theorem for complete self-shrinkers. As a corollary, we obtain if the Lagrangian graphM∇fis complete inRn2nand passes through the origin then it is flat.

Translating solutions to Lagrangian mean curvature flow

We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an L 2 L^2 bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee, and M.-P. Tsui shows that these conditions are optimal.

A gap theorem for translating solitons to Lagrangian mean curvature flow

In this paper, we prove a gap theorem for translating solitons to the almost calibrated Lagrangian mean curvature flow. More precisely, we prove that there exists a constant λ>0 such that if |H|2⩽λ|T|2, then the translating soliton must be a plane. We also obtain a similar result for symplectic mean curvature flow.

Dense gravity currents moving beneath progressive free-surface water waves

AbstractThe characteristics of dense gravity currents in coastal regions, where free-surface gravity waves are dominant, have yet to be studied in the laboratory. This paper provides a first insight into the dynamics of dense saline gravity currents moving beneath regular progressive free-surface water waves. The gravity currents were generated by releasing a finite volume of saline into a large wave tank with an established periodic wave field. After the initial collapse, the gravity currents propagated horizontally with two fronts, one propagating in the wave direction and the other against the wave direction. The fronts of the gravity currents oscillated with an amplitude and phase that correlated with the orbital velocities within a region close to the bed. To leading order, the overall length of the gravity current was found to be weakly affected by the wave action and the dynamics of the current could be approximated by simply considering the buoyancy of the released fluid. Other characteristics such as the position of the gravity current centre and the shape of the two leading profiles were found to be significantly affected by the wave action. The centre was displaced at constant speed dependent on the second-order wave-induced mean Lagrangian velocity. For long waves, the centre was advected downstream in the direction of wave propagation owing to the dominance of Stokes drift. For short waves, the gravity current centre moved upstream against the wave direction, as under these wave conditions Stokes drift is negligible at the bed. An asymmetry in the shape of the upstream and downstream current heads was observed, with the gravity current front moving against the waves being much thicker and the front steeper, similar to the case of a current moving in a stream.