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Parcel Eulerian–Lagrangian Fluid Dynamics of Rotating Geophysical Parcel Eulerian–Lagrangian fluid dynamics of rotating geophysical flows By Onno Bokhove1 and Marcel Oliver2 1Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands 2School of Engineering and Science, International University Bremen, 28759 Bremen, Germany Parcel Eulerian–Lagrangian Hamiltonian formulations have recently been used in structure-preserving numerical schemes, asymptotic calculations, and in alter- native explanations of fluid parcel (in)stabilities. A parcel formulation describes the dynamics of one fluid parcel with a Lagrangian kinetic energy but an Eulerian potential evaluated at the parcel’s position. In this paper, we derive the geometric link between the parcel Eulerian–Lagrangian formulation and well-known varia- tional and Hamiltonian formulations for three models of ideal and geophysical fluid flow: generalized two-dimensional vorticity-streamfunction dynamics, the rotating two-dimensional shallow water equations, and the rotating three-dimensional com- pressible Euler equations. Keywords: Eulerian and Lagrangian fluid dynamics; variational principles; Hamiltonian dynamics; rotating compressible Euler, shallow water and vortical flows 1. Introduction Conservation laws play an important role in geophysical fluid mechanics. In the ab- sence of forcing and dissipation, conservations laws can be derived systematically from the variational or Hamiltonian structure of the equations of fluid motion. Conservation or near-conservation of mass, energy, and vorticity in the underly- ing idealized model and its numerical treatment is considered desirable to enhance stability and accurate ensemble forecasting, even though the dynamics of the atmo- sphere and oceans are ultimately driven by forcing on the large scales and subject to viscosity on the small scales. Recently, Frank, Gottwald, and Reich (2002) and Frank and Reich (2003, 2004) introduced a Hamiltonian Particle Mesh (HPM) numerical method which conserves mass, energy, circulation, and phase space structure when a symplectic time integra- tion scheme is used. The HPM method is a parcel Eulerian–Lagrangian method: the fluid particles and velocities are discretized in a Lagrangian way, while the effective density is Eulerian and is discretized and smoothed on a fixed mesh. Smoothing of this density appears be required to ensure stability for time steps on the advection time scale. The conservation of mass and circulation in the HPM numerical model is shown by interpreting the dynamics of particles as a non-autonomous Hamiltonian system with particle position and particle velocity as variables; parcels interact via the Article submitted to Royal Society TEX Paper 2 Bokhove, Oliver transport of mass or vorticity (Frank and Reich, 2003). Generally, the Hamiltonian associated with a particle representation consists of a type of Bernoulli function, that is, the sum of the Lagrangian kinetic energy plus an Eulerian potential func- tion depending on space, evaluated at the position of the particle, and time. It turns out that in continuum fluid dynamics a single fluid parcel satisfies the same non-autonomous Hamiltonian formulation, which is finite dimensional for the one distinguished fluid parcel. The Eulerian potential is now transported by the flow generated by the parcel velocities, thereby coupling the parcels to a continuum. We call this the parcel Eulerian–Lagrangian formulation. The question we will answer is how such parcel Eulerian–Lagrangian formu- lations relate to the well-known Lagrangian or Eulerian continuum Hamiltonian mechanics. Three flow systems are considered in turn: generalized two-dimensional incompressible flow expressed in terms of stream function and (potential) vorticity, two-dimensional rotating shallow-water equations, and three-dimensional rotating compressible Euler equations for the motion of air. Besides the preservation of the Hamiltonian structure in the numerical HPM approach, the parcel formula- tion is often simpler than the more commonly used partial differential formula- tion in asymptotics (Bokhove, 2005), fluid parcel explanations of flow instabilities (Bokhove, 2005), and the proof of the Jacobi identity (this paper). In particular, the continuum Poisson bracket inherits the Jacobi identity by construction from the parcel formulation, where it is more easily verified. The link between the Hamilto- nian formulation and its parcel Eulerian–Lagrangian form for an ensemble of point vortices was already noted by Morrison (1981); we believe that the general case and the derivation through Lagrangian symmetries are new. The outline of the paper is as follows. In §2, we introduce definitions and re- lations between Lagrangian and Eulerian flows. An overview of the three parcel Eulerian–Lagrangian flows is given in §3. The relation between parcel and contin- uum dynamics is given in §4 to §6 in turn for the vortical flows, the shallow water equations, and the compressible equations for air. 2. Preliminaries The fundamental object in the continuum description of a fluid in a container D is the flow map χ: D × R → D which maps a fluid “parcel” labeled by its initial location a to its later Eulerian position x = χ(a, t). Depending on the application considered later, D may be two- or three-dimensional. We view χ(·, t) as a curve in the group of diffeomorphisms (or, if the system is incompressible, the subgroup of volume preserving diffeomorphisms) of D parameterized by t. The Lagrangian velocity χ˙ (a, t) is the velocity of the fluid parcel labeled a, while the Eulerian velocity u(x, t) is the velocity of the parcel passing location x at time t, so that, following the conventions of, for example, Marsden and Ratiu (1994; see also Oliver, 2006, for the shallow water case), χ˙ (a, t) ≡ ∂tχ(a, t) = u(χ(a, t), t) or χ˙ = u ◦ χ . (2.1) The effective density ρ(x, t), which may be the physical fluid density or, as in the case of the shallow water equations, the height function, is a density form on Article submitted to Royal Society Parcel Eulerian–Lagrangian fluid dynamics 3 which the flow map acts by pullback, i.e. ∗ χ ρ ≡ (det ∇χ) ρ ◦ χ = ρ0 , (2.2) where ρ0 = ρ0(a) is the initial effective density, which we often take equal to one for simplicity, and ∇χ = ∇aχ denotes the gradient with respect to the label coordinates; we will not explicitly denote the variables of differentiation when no confusion is possible. In the following, we will treat ρ as a scalar, so that ρ ◦ χ/ρ0 is the Jacobian determinant of the transformation; relation (2.2) can then be written in the equivalent integral form Z Z 0 0 0 ρ(x, t) = ρ(x , t) δ(x − x ) dx = ρ0(a) δ x − χ(a, t) da , (2.3) D D where δ(·) denotes the Dirac measure. A further formally equivalent relation is the Eulerian continuity equation ∂tρ + ∇ · (ρu) = 0 , (2.4) which follows from differentiating (2.2) in time and using (2.1). When the flow is incompressible, ∇ · u = 0; for simplicity, we will also assume ρ constant. If χε denotes a curve of flow maps parameterized by ε with χ0 = χ, the varia- tional derivative is defined as ∂χε δχ = . (2.5) ∂ε ε=0 It is useful to introduce a corresponding Eulerian variation w via δχ = w ◦ χ , (2.6) which can be seen as an analogy to χ˙ = u ◦ χ, so that u and w must satisfy the same boundary conditions. Moreover, just as (2.4) emerges from (2.1), we derive δρ + ∇ · (ρw) = 0 (2.7) from (2.6). For incompressible flows, this implies that ∇ · w = 0 so that, for any materially conserved quantity θ ◦ χ = θ0 ≡ Θ, δθ + w · ∇θ = 0 . (2.8) In the following, we localize various quantities to a single, distinguished fluid parcel, which we label A. Quantities relating to this distinguished parcel are also written in capital letters so that, in particular, X(t) = χ(A, t) and U(t) = χ˙ (A, t). We localize variations of the flow map by introducing the special variations δχ(a, t) = δ(a − A) δX (2.9) and δχ˙ (a, t) = δ(a − A) δU . (2.10) T ⊥ Finally, for vector fields u = (u1, u2) in two dimensions we write u = T (−u2, u1) and identify the curl with a scalar field, so that the curl operator in ⊥ two dimensions is conveniently written as ∇ · u = ∂1u2 − ∂2u1. Article submitted to Royal Society 4 Bokhove, Oliver 3. Parcel Eulerian–Lagrangian fluid dynamics (a) Two-dimensional generalized vorticity dynamics The generalized two-dimensional vorticity dynamics considered in Bernsen et al. (2006) take the parcel formulation dX = −B ∇⊥H = B ∇⊥ψ , (3.1a) dt v ∇ · (B∇ψ) − C ψ + D = ζ/B , (3.1b) Z ζ(X, t) = ζ0(a, t) δ X − χ(a, t) B χ(a, t) da , (3.1c) D where ζ = ζ(X, t) denotes the (potential) vorticity, ψ = ψ(X, t) the stream func- tion, u = B ∇⊥ψ denotes the parcel velocity which satisfies the weighted divergence condition ∇ · (B−1u) = 0, and χ denotes the flow generated by u. Equation (3.1c) expresses the material conservation of (potential) vorticity. The single parcel dy- namics is Hamiltonian with Hv(X, t) = −ψ(X, t) . (3.2) We assume that 0 < Bmin ≤ B(X) < ∞, C(X) ≥ 0 and D = D(X). For B = 1 and C = D = 0, the system reduces to the Euler equations for ideal incompressible flow. For other choices, the system reduces to the quasi-geostrophic or to the rigid lid equations with B−1 representing the bottom topography of the basin; for details see Bernsen et al. (2006). Boundary conditions depend on the setting, where the possible choices include doubly periodic, sufficient decay at infinity, and ψ = 0 with possibly circulation conditions if the domain is not simply connected. In the simplified case that the Hamiltonian Hv(X, t) is prescribed, as is com- monly done in studies of chaotic advection, the Hamilton equations (3.1a) form a kinematic description.
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