Geometrical phases, such as the Berry phase, have proven to be powerful concepts to understand numerous physical phenomena, from the precession of the Foucault pendulum to the quantum Hall effect and the existence of topological insulators. The Berry phase is generated by a quantity named the Berry curvature, which describes the local geometry of wave polarization relations and is known to appear in the equations of motion of multi-component wave packets. Such a geometrical contribution in ray propagation of vectorial fields has been observed in condensed matter, optics and cold atom physics. Here, we use a variational method with a vectorial Wentzel–Kramers–Brillouin ansatz to derive ray- tracing equations for geophysical waves and to reveal the contribution of the Berry curvature. We detail the case of shallow-water wave packets and propose a new interpretation of their oscillating motion around the equator. Our result shows a mismatch with the textbook scalar approach for ray tracing, by predicting a larger eastward velocity for Poincaré wave packets. This work enlightens the role of the geometry of wave polarization in various geophysical and astrophysical fluid waves, beyond the shallow-water model.

1. Introduction

The concept of a geometric phase provides a unified framework to interpret physical phenomena as diverse as the Foucault pendulum experiment [1] or the Aharonov–Bohm effect [2]. It arises in virtually all fields of physics, including geophysical systems [3–5]. These geometrical phases are the manifestation of an underlying geometrical property of the system, called the curvature F, such that the phase ϕ is formally related to it by an integral formula over a surface as $ϕ = (1 / 2 π) \int F d S$ .

For instance, for the Foucault pendulum, the geometrical phase corresponds to its precession angle, called the Hannay angle, that results from the Earth’s curvature. Here, we are concerned with multi-component wave problems which generate a more abstract curvature, called the Berry curvature. Such a curvature encodes the polarization relations between the different wave components in parameter space. Although it was originally proposed in quantum physics [2], the emergence of a Berry curvature only requires the existence of several wave bands: each band n is then characterized by a Berry curvature F⁽ⁿ⁾. As such, it appears naturally in multi-component wave problems, from quantum condensed matter systems [6,7] to classical waves in optics [8,9], plasma physics, mechanics [10,11] and hydrodynamics [12,13]. A direct physical manifestation of the Berry curvature itself—rather than the phase it is related to—arises in ray-tracing experiments: simply put, the Berry curvature is a correction of the group velocity that bends ray trajectories [6,14]. This correction was measured in condensed matter [15,16], cold atom set-ups [17] and photonic quantum walks [18]. Although geophysical flows gather all the ingredients for a Berry curvature to emerge, it has up to now not been established how this curvature arises in geophysical ray tracing. This is the purpose of the present article.

Previous works have shown that inertia-gravity wave (or Poincaré wave) bands of the rotating shallow-water model admit a non-zero Berry curvature [12]. This is therefore a natural starting point to address possible deviations of ray trajectories, which we derive using a variational principle inspired by quantum mechanics [14,19]. More generally, we propose a consistent framework to reconsider the ray-tracing problem in a large class of multi-component geophysical flow models, building upon previous results on vectorial elliptical linear systems [20–22]. This new approach emphasizes the role of Berry curvature effects that were up to now overlooked in geophysics.

To be more specific, we consider the dynamics of a wave packet in a medium characterized by a length ℓ, varying spatially over a typical distance L. To fix the ideas, we will consider as our main example shallow-water wave packets evolving in a fluid layer. In that case ℓ is called the Rossby radius of deformation, which is inversely proportional to the rotation rate of the frame of reference [23].

The wave packet has a mean wavelength λ and is modulated by an envelope of typical size Λ > λ. We assume that the extension of the wave’s envelope Λ is much smaller than the typical distance L depicting the medium’s variations (λ, Λ ≪ L). This amounts to assuming that the solution is localized regarding the scale of the medium’s variations, thus allowing one to project the packet onto a plane-wave solution with wavelength λ and local ℓ as parameters. We restrict ourselves to cases where the medium’s properties enter the dispersion relation at lowest order, which, in the family of models considered in this paper, adds up to assuming ℓ ∼ λ. To sum up, the lengths describing the problem and its wave packet solution are organized as follows:

ℓ \sim λ < Λ ≪ L .

1.1

The usual way to derive ray-tracing equations in geophysics is using a Wentzel–Kramers–Brillouin (WKB) ansatz [24–26] under scaling conditions such as (1.1). At the scale of the wavelength, the medium is approximately homogeneous and steady, which can be quantified by the introduction of a small dimensionless parameter, say ε = λ/L. This scale separation justifies the use of a WKB ansatz for a solution described by a scalar field v,

v (r, t) = e^{i [(Φ_{0} / ε) + Φ_{1} + O (ε)]} .

1.2

Along with slowly varying (in time and space) functions Φ_i of order 1, expression (1.2) corresponds to the idea of a plane wave modulated by an envelope of finite size, with an identifiable mean wavevector. In geophysics, a WKB ansatz is indeed generally injected in a scalar equation as if the problem were entirely defined by one scalar equation and a sufficient number of boundary conditions [23,25,26]. Nevertheless, what about the models that are not well defined by and reducible to just one equation, even of higher order, and rather by a set of coupled wave equations with distinct coefficients? Such a set of equations can be rewritten as one with a multi-component field, as will be illustrated in the following, which makes the WKB approximation more complex. Indeed a multi-component mode is characterized by both the dispersion and the polarization relations, the latter relating the different component fields between each other, all depending on the wavevector and the medium’s parameters. Therefore, as the wave packet moves through the slowly varying medium, its polarization relations change accordingly as the ray coordinates (r, k) evolve. As a consequence, we expect a footprint of the corresponding Berry curvature in the ray dynamics of multi-component geophysical wave problems.

The shallow-water model with Coriolis parameter f varying with meridional coordinate y is a classical three-component wave model that cannot be reduced to one scalar equation. It will be used as the key example throughout this paper. A scalar equation for the meridional component of the velocity field v(x, y, t) alone can be derived (see §4.c), then one can inject ansatz (1.2) into this scalar equation in order to infer an approximate WKB solution for v and find the corresponding relations for the phase functions at different orders. The results obtained via this approach for ray tracing can be found in various articles such as [27–29] and textbooks [23,30], and are recalled in §4.c. However, the technique suffers from two difficulties.

—	First, the field v alone is not sufficient to fully characterize the energy propagation. Indeed, the two other fields involved in the shallow-water dynamics do not satisfy the same wave equation as v when the Coriolis parameter is not homogeneous. Instead, they are both coupled to the field v. This leads to inconsistent WKB ansatzes for the different fields of the problem, as noted by Godin [4] in the context of compressible stratified waves. In fact, keeping just a single scalar wave equation amounts to only considering the dispersion relation and losing all the information on the polarization relations between the fields. In other words, the scalar approach misses any geometrical effect related to the polarization relations of the eigenmodes, which is an essential feature in the presence of inhomogeneity. Thereby the geometrical nature of the problem must be accounted for.
—	Second, and just as importantly, we find that the usual textbook derivation of ray-tracing equations from a scalar equation involves a contradictory hypothesis on the scale of variation of the Coriolis parameter: keeping the variations of the Coriolis parameter at leading order of the WKB expansion (which is the dispersion relation that normally accounts for the homogeneous parameters) amounts to assuming that it varies considerably over a distance λ (typically a few kilometres for geophysical shallow-water waves on Earth), whereas it does vary over a scale L that is much greater than λ (L is typically 1000 km on Earth). We show that it is therefore necessary to reconsider the order of magnitude of a small parameter characterizing the inhomogeneity of the medium in order to properly derive ray-tracing equations at order 1 in this small parameter.

The issue of solving multi-component differential systems has long been addressed in applied mathematics [20,21,31,32]. This has been related to the existence of geometric phases in the vectorial WKB solutions of multi-component wave problems [3,4,22,33,34]. However, the role of Berry curvature in ray tracing—which is a simpler problem because it does not aim at computing precisely the evolution of the phase—has not been shown in the geophysical context. The purpose of the present work is to rectify the situation.

The paper is organized as follows. In §2., we introduce the shallow-water model and its notations, the dimensionless parameters characterizing it and the spectral properties of the homogeneous problem, which are necessary to construct a wave packet solution for the slowly varying medium. Then in §3., we present a multi-component ansatz and a variational formulation of the problem, accounting for the different orders of the problem in a more consistent way and leading to the set of ray-tracing equations. We show that the ray motion of inertia-gravity waves involves the Berry curvature and, as a by-product, we report a direct analogy between the behaviour of inertia-gravity wave rays on β-plane Earth and the anomalous Hall effect [6,35]. Finally, in §4., we compare the result with the traditional scalar WKB approach for ray tracing.

2. Formulation of the problem for shallow-water waves

To illustrate the general issue of ray tracing for multi-component wave problems in a geophysical setting, we propose to treat in this section the particular case of energy propagation through surface waves described by the shallow-water model, keeping in mind that the results presented in §3. have a more general extent.

(a) The linearized rotating shallow-water model

Let us consider the shallow-water model in planar geometry, describing a thin layer of fluid with homogeneous density in the presence of gravity (figure 1). It is a bidimensional model, so the position reads $r \equiv x {\hat{e}}_{x} + y {\hat{e}}_{y}$ and we write the planar velocity field $u \equiv u {\hat{e}}_{x} + v {\hat{e}}_{y}$ . For the sake of simplicity, we focus on the case of a flat bottom, with a constant phase speed denoted c for surface waves. We consider a version of the model linearized around a state at rest u ≡ 0 in the rotating frame of reference. We also use here the traditional approximation that consists in only considering the components of the Coriolis force in the horizontal plane, through the Coriolis parameter f, which is twice the projection of the planet’s angular velocity vector on the local plane’s vertical axis (figure 1). All fields, parameters and variables of the problem are non-dimensionalized with length λ and time λ/c. Then the shallow-water equations governing the dynamics of the perturbation fields of velocity u, v and elevation or geopotential η, all much smaller than 1 in amplitude, read

\begin{aligned} \partial_{t} u = - \partial_{x} η + f (y) v, \end{aligned}

2.1a

\begin{aligned} \partial_{t} v = - \partial_{y} η - f (y) u \end{aligned}

2.1b

\begin{aligned} and & \partial_{t} η = - \partial_{x} u - \partial_{y} v . \end{aligned}

2.1c

Equations (2.1a) and (2.1b) are the projections of the linearized momentum conservation equation while (2.1c) comes from the conservation of mass. Note that the only varying parameter in this problem is the adimensionalized Coriolis parameter f, which is central because it breaks the time-reversal symmetry of equations (2.1) and opens a gap in their dispersion relation (figure 2). Equations (2.1) describe the free solutions of Laplace’s tidal equations, in Cartesian coordinates. The adimensionalized length

ℓ = \frac{1}{| f |}

2.2

is the Rossby radius of deformation, and the gradient of f with latitude is usually denoted by β.

Figure 1. (a) The shallow-water model describes wave propagation on the free surface of a thin layer of homogeneous fluid, whose total elevation is a function of time and position and reads h ≡ H(1 + η). The mean depth H must be much smaller than the typical horizontal variation scales, i.e. H ≪ λ with λ the typical horizontal wavelength. In that case $c = \sqrt{g H}$ . (b) Planar approximation: the model is applied to a flow domain that takes place on a locally tangent plane on the planet, with local vertical basis vector $\hat{n}$ —with Ω the rotation vector of the Earth, the Coriolis parameter is defined as $f = 2 Ω \cdot \hat{n}$ . (Online version in colour.)

Figure 2. Plot of the dispersion relation of the f-plane shallow-water equations for the three bands (indexed by n = −1, 0, + 1). The gap between them equals |f|. The colour bar shows the value of the component $F_{k_{x} k_{y}}$ of the Berry curvature, defined in equation (2.11), as a function of k and for a fixed positive value of f, for each band. (Online version in colour.)

(b) Dimensionless parameters of the problem and slow variables

We consider the problem of a shallow-water wave packet evolving in a medium with a varying Coriolis parameter. The dynamics is given by the linear rotating shallow-water equations (2.1) and a typical initial condition as depicted by relations (1.1). The problem involves four length scales, and admits therefore three non-dimensional parameters,

f, ε \equiv \frac{λ}{L} and α \equiv \frac{λ}{Λ} .

2.3

Let us comment on each of them:

—	f ∼ 1 is a function of the slow variable Y ≡ εy, varying over a typical scale Y ∼ 1. We indiscriminately write f as a function of y or Y, as the case may be.
—	ε ≪ 1 is the inverse of the adimensionalized typical length for the Coriolis parameter’s variation with latitude. It is the control parameter in the WKB expansion. The contribution of the Berry curvature to ray-tracing equations arises as a correction of order ε, as will be shown in §3.
—	$α ≲ 1$ reflects the ratio between the wavelength and the spatial extension of the packet. For the wave packet to behave as a plane wave at leading order, its envelope’s extension must be at least a few times the wavelength, and thus α must be smaller than 1. The ratio ε/α = Λ/L appears as a small control parameter that guarantees that the medium is quasi-homogeneous at the scale of the wave packet. Therefore, α must not be too small (i.e. of the order of ε), otherwise this localized wave packet picture fails.

We propose to organize these parameters according to the set of inequalities $ε ≪ α^{2} ≲ 1$ to derive ray-tracing equations, together with f(Y) ∼ 1. This last choice guarantees that rotation enters at lowest order in the wave dispersion relation. The reason for the scaling ε ≪ α² will be clarified in §3.a.

Consequently, we suppose that the phase functions are natural functions of the slow variables

(X, Y, T) \equiv (R, T) \equiv (ε x, ε y, ε t),

2.4

whereas the amplitude function is by definition a natural function of the variables

(X^{'}, Y^{'}, T^{'}) \equiv (R^{'}, T^{'}) \equiv (α x, α y, α t) .

2.5

Therefore, the existence of these two parameters describing the solution leads to a multiscale characterization of the ansatz. In the following, we sometimes use instead the derivation operators with respect to these coordinates, namely

\nabla_{R} \equiv (\partial_{X}, \partial_{Y}) and \nabla_{R^{'}} \equiv (\partial_{X^{'}}, \partial_{Y^{'}}) .

2.6

Introducing these notations does not mean that we aim at computing the evolution of the wave packet’s phase, which is not relevant in this ray-tracing context. Our assumptions, however, allow us to treat the variation of the Coriolis parameter as a perturbation—as it should be, considering its order of magnitude—that will enter ray-tracing equations at lower order. As explained in the Introduction, we need such geometric tools as the Berry curvature and a vectorial WKB ansatz in order to address this perturbation correctly regarding the multi-component nature of shallow-water waves.

(c) Spectral and geometrical features of waves in the f-plane model

Following [26], in order to construct the wave packet solution for waves in a slowly varying medium we must first and foremost derive the plane-wave solutions supported by the translationally invariant medium. Equations (2.1) can be rewritten in a more compact Schrödinger-like equation

\begin{aligned} (i \partial_{t} - \hat{H} (r, \nabla)) ψ (r, t) = 0, \end{aligned}

2.7

\begin{aligned} with & \hat{H} (r, \nabla) \equiv (\begin{matrix} 0 & i f (y) & - i \partial_{x} \\ - i f (y) & 0 & - i \partial_{y} \\ - i \partial_{x} & - i \partial_{y} & 0 \end{matrix}), ψ (r, t) \equiv (\begin{matrix} u \\ v \\ η \end{matrix}), \end{aligned}

2.8

where

\hat{H} (r, \nabla)

is a linear differential operator and ψ(r, t) is a three-component vector field, called wave function hereafter.

It is instructive to consider the simplest possible case where the Coriolis parameter does not vary in space, the celebrated f-plane approximation (figure 1), originally studied by Kelvin [36]. Plane-wave solutions are then allowed and written in the form ψ(r, t) = e^{i(k·r−ωt)} U_k,ω. The problem (2.7) boils down to a matricial equation

H [f, k] U_{k, ω} = ω U_{k, ω}, with H [f, k] \equiv (\begin{matrix} 0 & i f & k_{x} \\ - i f & 0 & k_{y} \\ k_{x} & k_{y} & 0 \end{matrix}),

2.9

where

U_{k, ω} \equiv {(\begin{matrix} {\hat{u}}_{k, ω} {\hat{v}}_{k, ω} {\hat{η}}_{k, ω} \end{matrix})}^{t} \in C^{3}

is the vector of Fourier transforms and k ≡ (k_x, k_y). Normalized eigenmodes and eigenvalues of this problem are denoted, respectively, ω_n[f, k] and U_n[f, k], with n ∈ { − 1, 0, 1} for the shallow-water model. In this particular case, one gets

ω_{0} = 0, ω_{\pm 1} = \pm \sqrt{f^{2} + k^{2}} .

2.10

The first wave is known as the geostrophic mode and the second as the inertia-gravity or Poincaré wave. The dispersion relation is depicted in figure 2. Polarization relations of the waves are given by the eigenmodes U_n[f, k], which are provided in appendix A.

The normalized eigenmodes U_n are parameterized over a base space (f, k_x, k_y). At each point of this base space, each vector U_n is defined up to a phase, which defines a complex eigenspace of dimension 1. The continuous family of such parametrized eigenspaces over (f, k_x, k_y) defines a fibre bundle, or eigenbundle, which owns specific geometric properties, whose repercussions for the wave packet dynamics is the central motivation of this work. Those properties can be quantified by the Berry curvature, defined as

F_{λ_{μ} λ_{ν}}^{(n)} = i (\frac{\partial U_{n}^{†}}{\partial λ_{μ}} \frac{\partial U_{n}}{\partial λ_{ν}} - \frac{\partial U_{n}^{†}}{\partial λ_{ν}} \frac{\partial U_{n}}{\partial λ_{μ}}),

2.11

where the

λ_{μ}

are the coordinates of the base space, e.g. λ ≡ (f, k_x, k_y) here. Importantly, this quantity is a gauge-invariant real-valued quantity: it does not depend on the phase choice for the eigenmodes U_n. Qualitatively, it describes local twisting of these eigenmodes. It was computed for shallow-water waves in the f-plane, i.e. for a fixed value of f, in [12]. The value of

F_{k_{x} k_{y}}

for f-plane waves is reproduced with the dispersion relation in figure 2.

Let us now turn to a more general case of rotating shallow-water waves where f(r) varies spatially. It is useful to consider the symbol H[f(r), k], which in our case is formally defined by replacing $\nabla$ in the operator $\hat{H} (r, \nabla)$ of (2.8) by ik, as in equation (2.9) for shallow-water equations. Eigenmodes and eigenvalues of the symbol are just those of the f-plane problem. The study of symbols is at the heart of a mathematical framework called the Weyl calculus [31]. It provides a powerful tool to study wave problems in physics, starting from semi-classical analysis (e.g. [20,21,37]). The strong interest of this approach in geophysical wave transport has recently been advocated by Onuki [22]. We show in §3. how the Berry curvature emerges in this framework.

3. Berry curvature in ray tracing of multi-component waves

In this section, we derive ray-tracing equations by exploiting the equivalence between the Schrödinger-like equation (2.7) and a problem of functional optimization [14,19]. First, we introduce a vectorial ansatz for the wave packet, then we show that the Lagrangian describing its dynamics involves the Berry connection of the wave’s eigenspace. The results of this section are not specific to the shallow-water model, and can be applied to any other continuous multi-component wave system provided that the scaling discussed in §2.b is respected.

(a) A ray-tracing method based on a variational principle for multi-component wave problems

Our starting point to address the Schrödinger-like equation (2.7) is the following vectorial ansatz:

ψ^{'} (r, t) = A (α r, α t) e^{i Φ (ε r, ε t)} U_{n} [f (ε y), k_{c} (ε t)] .

3.1

This ansatz corresponds to the initial condition discussed in the Introduction, which is a wave packet on band n with an adiabatically varying mean wavevector k_c(T) and corresponding local eigenmode U_n [f(Y), k_c(T)], modulated by the envelope function A(X′, Y′, T′) (with the notations introduced in relations (2.4) and (2.5) for the variables X, Y, T, X′, Y′ and T′). Again, all functions in ansatz (3.1) can be written as a function of the slow or normal variables in the following, as the case may be. Both the slowly varying phase Φ(X, Y, T) and the envelope function A of the vectorial field are real-valued, contrary to the scalar ansatz (1.2), where the phase was given by $R e Φ$ and the amplitude by $\exp (- I m Φ)$ , which is more convenient for the scalar method. We can expand Φ = Φ₀/ε + Φ₁ + · · ·, with all Φ_i of order 1, in a WKB framework, but we do not expand further than i = 1, all the geometrical effects of interest here being expected to appear at this order. One needs to eventually take the real part of the wave function (3.1) to find the actual solution of the fluid problem. It is however more convenient to work with the complex solutions throughout this paper.

We aim at deriving the ray-tracing equations, meaning the equations of motion of the coordinates defining the wave packet: the position r_c(t) and momentum k_c(t) of its centre of mass, which we hereby define. It is convenient to use a bracket notation, although with non-normalized vectors. For instance, by definition of the canonical inner product of the Hilbert space $L_{2} (R^{2}, C^{3})$ ,

⟨ ψ (t) | ψ (t) ⟩ \equiv \iint d x d y ψ {(x, y, t)}^{†} ψ (x, y, t) = \int d^{2} r A {(α r, α t)}^{2},

3.2

the adimensionalized mechanical energy of the perturbation reads

E = \frac{1}{2} \iint d x d y (| u (x, y, t) |^{2} + | v (x, y, t) |^{2} + | η (x, y, t) |^{2}) = \frac{1}{2} ⟨ ψ | ψ ⟩ .

3.3

The energy is a constant of motion since the operator

\hat{H}

is Hermitian. Similarly, the position and momentum of the centre of the wave packet are defined with Hermitian operators as

r_{c} \equiv \frac{⟨ ψ | r | ψ ⟩}{⟨ ψ | ψ ⟩} and k_{c} \equiv \frac{⟨ ψ | (- i \nabla) | ψ ⟩}{⟨ ψ | ψ ⟩} .

3.4

The latter must be equal to the wavevector preassigned in ansatz (3.1). This consistency condition yields the following relation, shown in appendix B:

k_{c} (t) = \nabla_{R} Φ_{0} + ε (\nabla_{R} Φ_{1} - i U_{n}^{†} \nabla_{R} U_{n}) + O ({(\frac{ε}{α})}^{2}),

3.5

all the phase functions in relation (3.5) being evaluated at time t and position r_c(t), eigenmode U_n at [f(Y_c(T)), k_c(T)].

Solving equation (2.7) is equivalent to the variational problem of finding the function ψ′ (r, t) optimizing the action integrated from the following Lagrangian [14,19]:

L^{'} [ψ^{'}] \equiv \frac{⟨ ψ^{'} | (i \partial_{t} - \hat{H} (r, \nabla)) | ψ^{'} ⟩}{⟨ ψ^{'} | ψ^{'} ⟩} .

3.6

The solution of the variational problem (3.6) directly gives a solution of equation (2.7), differing by a time-dependent phase that does not change any of the observables defined by relations (3.3) and (3.4) [19].

This variational approach is convenient for ray tracing because the Lagrangian $L^{'}$ defined by expression (3.6) depends only on the variables of centre of mass r_c and k_c and their time derivatives at lowest orders (see appendix B for the detailed derivation),

L^{'} = {\dot{r}}_{c} \cdot k_{c} + {\dot{r}}_{c} \cdot i U_{n}^{†} \frac{\partial U_{n}}{\partial r_{c}} + {\dot{k}}_{c} \cdot i U_{n}^{†} \frac{\partial U_{n}}{\partial k_{c}} - Ω^{(n)} (r_{c}, k_{c}) - \frac{d}{d t} χ (t) + O ({(\frac{ε}{α})}^{2}),

3.7

with the notations

\dot{X} \equiv d X / d t

and

U_{n} (r_{c}, k_{c}) \equiv U_{n} [f (r_{c}), k_{c}] and Ω^{(n)} (r_{c}, k_{c}) \equiv \frac{⟨ ψ | \hat{H} | ψ ⟩}{⟨ ψ | ψ ⟩} + O ({(\frac{ε}{α})}^{2}),

3.8

where |ψ〉 is the wave function and index n indicates the band over which the packet is defined, as written in expression (3.1). In other words, defining the reduced Lagrangian as

L (r_{c}, k_{c}, {\dot{r}}_{c}, {\dot{k}}_{c}) \equiv {\dot{r}}_{c} \cdot k_{c} + {\dot{r}}_{c} \cdot i U_{n}^{†} \frac{\partial U_{n}}{\partial r_{c}} + {\dot{k}}_{c} \cdot i U_{n}^{†} \frac{\partial U_{n}}{\partial k_{c}} - Ω^{(n)} (r_{c}, k_{c}),

3.9

the difference between functionals

L^{'}

and

L

adds up to second-order terms and the exact time derivative of a function χ(t) that can be removed, as it does not affect the solutions of the variational problem (see appendix B).

We recognize in expression (3.9) for $L$ the Berry connection $i U_{n}^{†} d U_{n}$ of the nth eigenmode’s bundle parametrized over the ray phase space (r_c, k_c) as the base space (see §2.c), with

{\dot{r}}_{c} \cdot i U_{n}^{†} \frac{\partial U_{n}}{\partial r_{c}} + {\dot{k}}_{c} \cdot i U_{n}^{†} \frac{\partial U_{n}}{\partial k_{c}} = i U_{n}^{†} \frac{d U_{n}}{d t} .

3.10

We explain in §3b the relation between this function and the Berry curvature introduced in §2. This quantity connects U_n (r_c + dr_c, k_c + dk_c) to U_n (r_c, k_c), thus providing a footprint of the vectorial polarization relations in the ray trajectory in phase space. If the connection were null we would recover the simple form

L = {\dot{r}}_{c} \cdot k_{c} - Ω^{(n)} (r_{c}, k_{c})

and simply identify observables r_c, k_c and Ω⁽ⁿ⁾—defined in equations (3.4) and (3.8)—to the generalized position, momentum¹ and Hamiltonian² of classical analytical mechanics, respectively. Such a trivial case would occur for instance in a one-band system or, more generally, in the case of a band with topologically flat eigenbundles: all Berry connections, of the form

i U_{n}^{†} \partial U_{n} / \partial q

as appearing in expression (3.10), are null in that situation.

Expression (3.9) justifies the interest of this variational principle altogether with the approximation of a wave packet in a slowly varying medium: the dynamics of such a solution can be reduced to that of its coordinates of the centre of mass. The results of this section do not depend on the choice of ansatz (3.1), as long as the scaling $ε ≪ α^{2} ≲ 1$ , introduced in §2.b, is respected. More precisely, the reduced Lagrangian $L$ defined in equation (3.9) and the full Lagrangian $L^{'}$ defined in equation (3.6) are equal up to order ε in the regime (ε/α)² ≪ ε ≪ 1.

(b) Ray-tracing equations and the Berry curvature

The coordinates of the wave packet in phase space, r_c(t) and k_c(t), are the generalized coordinates of the system whose Lagrangian $L$ is defined by equation (3.9), and their time derivatives are the generalized velocities. Cancelling the lowest-order variations of the action integrated from $L$ yields the Euler–Lagrange equations,

\frac{\partial L}{\partial q} = \frac{d}{d t} (\frac{\partial L}{\partial \dot{q}}),

3.11

where q is any of the r_c or k_c components. Coordinates x and y will be denoted by Greek letters μ or ν for Einstein tensorial notations in the following. The Berry connection arising in expression (3.9) results in the conjugated momentum of r_c being equal to k_c plus an additional geometric term related to the Berry connection,

\frac{\partial L}{\partial {\dot{r}}_{c}} = k_{c} + i U_{n}^{†} \frac{\partial U_{n}}{\partial r_{c}} and \frac{\partial L}{\partial {\dot{k}}_{c}} = i U_{n}^{†} \frac{\partial U_{n}}{\partial k_{c}},

3.12

for the conjugated momentum of k_c. Deriving expression (3.9) and injecting expressions (3.12) in the Euler–Lagrange equations (3.11) finally yields, with q = k_c and q = r_c, respectively,

{\dot{r}}_{μ c} = + \frac{\partial Ω^{(n)}}{\partial k_{μ c}} - F_{k_{μ c} r_{ν c}}^{(n)} {\dot{r}}_{ν c} - F_{k_{μ c} k_{ν c}}^{(n)} {\dot{k}}_{ν c}

3.13a

and

{\dot{k}}_{μ c} = - \frac{\partial Ω^{(n)}}{\partial r_{μ c}} + F_{r_{μ c} r_{ν c}}^{(n)} {\dot{r}}_{ν c} + F_{r_{μ c} k_{ν c}}^{(n)} {\dot{k}}_{ν c} .

3.13b

The Berry curvature tensor F⁽ⁿ⁾ is defined as in equation (2.11), with

λ_{μ} \in {x_{c}, y_{c}, k_{x c}, k_{y c}}

for the shallow-water model, choosing the phase-space coordinates to parametrize the eigenmodes U_n(λ) ≡ U_n(r_c, k_c) of the symbol H(λ) ≡ H[f(r_c), k_c], with eigenvalue ω_n(r_c, k_c) ≡ ω_n[f(r_c), k_c]. Since U_n is normalized for every (r_c, k_c), the Berry connection

i U_{n}^{†} d U_{n}

has real coefficients.

As shown in [2], the Berry curvature of the nth wave bundle defined in equation (2.11) can be expressed with off-diagonal elements of the symbol’s derivatives,

F_{λ_{μ} λ_{ν}}^{(n)} = - 2 I m (\sum_{m \neq n} \frac{U_{n}^{†} (\partial H / \partial λ_{μ}) U_{m} U_{m}^{†} (\partial H / \partial λ_{ν}) U_{n}}{{(ω_{n} - ω_{m})}^{2}}) (Berry curvature) .

3.14

As for Ω⁽ⁿ⁾, defined by relation (3.8), we show the following expression in appendix B:

Ω^{(n)} - ω_{n} = - I m (\sum_{m \neq n} \frac{U_{n}^{†} (\partial H / \partial r_{μ c}) U_{m} U_{m}^{†} (\partial H / \partial k_{μ c}) U_{n}}{ω_{n} - ω_{m}}) (gradient correction) .

3.15

The gradient correction Ω⁽ⁿ⁾ − ω_n comes from the spatial variations of the medium’s parameters: it is a dynamical quantity. Conversely, the curvature F⁽ⁿ⁾ as expressed in expressions (2.11) or (3.14) only depends on the choice made to parametrize the manifold over which the eigenmodes are defined: it is a purely geometric quantity. As a by-product, Ω⁽ⁿ⁾ − ω_n is exactly null for the homogeneous problem, contrary to the Berry curvature F⁽ⁿ⁾. Both the ray Hamiltonian and the Berry curvature corrections in (3.13) are of order ε at best; therefore, we recover the classical scalar ray-tracing equations

{\dot{r}}_{c} = \partial ω_{n} / \partial k_{c}

and

{\dot{k}}_{c} = - \partial ω_{n} / \partial r_{c}

at leading order, with ω_n given by the dispersion relation.

Expressions (3.14) and (3.15) give, respectively, the curvature and the ray Hamiltonian with interband terms m ≠ n, showing that the corrections are larger at points k where the band n is close to another one in frequency and vanish as the bands are separated in frequency.³ This effect is visible in figure 2 in the shallow-water case, with higher values of $F_{k_{x} k_{y}}$ when the triplet (f, k_x, k_y) becomes closer to the degeneracy point (0, 0, 0). We shall keep in mind that ω_n, ω_m, Ω⁽ⁿ⁾, H, U_n and U_m in these relations are all functions of (r_c, k_c) = (x_c, y_c, k_xc, k_yc).

The set of equations (3.13) is different from the ones obtained with the traditional scalar derivation (see §4.c) as it accounts for both the corrected ray Hamiltonian Ω⁽ⁿ⁾ and the geometry of the wave packet’s polarization relations in ray phase space through the Berry curvature. Such a form of ray-tracing equations has already been derived for different purposes such as the study of the time-dependent Schrödinger equation in the adiabatic limit [19], the theory of Bohr–Sommerfeld quantization [37] and semiclassical analyses of wave packet trajectories in slowly perturbed crystals [6,14,40,41], optical lattices [18,42] and ultracold atoms [17]. Formal expressions for first-order corrections to transport equations in multi-component fluid wave problems with inhomogeneous media were proposed by Onuki [22], with concrete applications to shallow-water waves. However, the explicit contribution of the Berry curvature to ray-tracing equations has never been exhibited in astrophysical or geophysical fluid dynamics. Since a non-trivial Berry curvature has recently been reported in magnetized plasma [43] and compressible stratified fluids [13], we expect that geometrical corrections to ray-tracing equations will matter in those contexts. For now, we illustrate this effect by focusing on shallow-water wave rays with varying background rotation.

4. Application to shallow-water wave packets

(a) β-plane shallow-water waves

As shown in §3., the ray equations of motions (3.13) require only the knowledge of the ray Hamiltonian Ω⁽ⁿ⁾ (r_c, k_c) and the Berry curvature tensor F⁽ⁿ⁾ (r_c, k_c), which is a 4-by-4 matrix in the particular case of shallow-water equations. In other words, with equations (3.13) we are dealing with a double correction to the usual ray-tracing equations. These additional terms appear at lowest order in the medium’s variations, since their expressions only involve space derivatives of the symbol H[f(r_c), k_c] and its eigenmodes. We will therefore only need gradient corrections of the Coriolis parameter to apply equations (3.13) to shallow-water wave rays. It is therefore useful to introduce a new quantity,

β (y) \equiv \frac{d f (y)}{d y} = ε \frac{d f (Y)}{d Y} .

4.1

By definition β is of order ε. When β is assumed to be constant, we recover the β-plane approximation, usually exploited to model wave propagation in the equatorial area [12,27–29,44,45]. Note that the ray Hamiltonian Ω⁽ⁿ⁾ for shallow-water waves is exactly equal to ω_n in the f-plane approximation (β = 0) but in that case the tensor F⁽ⁿ⁾ is not null.

Just like traditional ray-tracing methods (based on the eikonal equation [26,45] or another scalar equation [23,27–30]), we just need to know the properties of the homogeneous (f-plane) system to infer the leading-order deviations arising from inhomogeneities (β-plane). The upgrade brought by the variational method presented in §3. is that it adds the f-plane polarization relations (hence the Berry curvature) to the dispersion relation of the wave, thus accounting for its vectorial character. Then from the homogeneous model’s eigenvalues ω_n and eigenmodes U_n, we just need to compute Ω⁽ⁿ⁾, defined by expression (3.15), and F⁽ⁿ⁾, defined by expression (3.14), and inject them into equations (3.13) to infer the trajectory of the corresponding wave packets (figure 3). We consider the shallow-water model’s different bands n: geostrophic waves n = 0 and Poincaré waves n = ±1. Since the shallow-water equations are time-invariant, the ray Hamiltonian Ω⁽ⁿ⁾ (r_c(t), k_c(t)) is a ray-tracing constant of motion, which can be shown directly from the antisymmetry of equations (3.13).

Figure 3. Simulation of an inertia-gravity wave packet in the Northern Hemisphere, computed with Dedalus [46]. The initial wavelength is 1, f₀ = 3 is the Coriolis parameter at the middle ordinate of the frame, β = 0.6 and the simulation runs until t = 32. (a) The red dots are the position of the centre of mass r_c(t) computed with the fields—from definition (3.4)—at regular steps of the simulation. (b) We zoom around the last position of the packet and compare the real trajectory (red dots) with the ones predicted, respectively, by the geometric ray-tracing equations (4.3) (red curve), scalar equations (4.9) (blue curve) and elementary ray-tracing equations (4.4) (green curve). Although the difference still appears small after t = 32, the geometric theory fits best the actual trajectory. (Online version in colour.)

(b) Berry curvature contribution to eastward drift of inertia-gravity waves

Let us now compute the drift of shallow-water vectorial wave packets induced by first-order corrections to ray-tracing equations, for the different bands n = −1, 0, 1 successively, by computing the observables Ω⁽ⁿ⁾ and F⁽ⁿ⁾.

The connection $i U_{0}^{†} d U_{0}$ of the geostrophic band n = 0 is null everywhere, hence so is its Berry curvature F⁽⁰⁾ (see appendix A). Therefore, only the dynamical correction remains in the set of equations of motion (3.13). Expression (3.15) with n = 0 yields

Ω^{(0)} (r_{c}, k_{c}) = - \frac{β k_{x c}}{f {(y_{c})}^{2} + k_{c}^{2}},

4.2

which is nothing more than the dispersion relation of quasi-geostrophic planetary or Rossby waves [23]. In that case the Berry curvature terms in equations (3.13) disappear and we simply retrieve Hamilton’s equations

{\dot{r}}_{c} = \partial Ω^{(0)} / \partial k_{c}

and

{\dot{k}}_{c} = - \partial Ω^{(0)} / \partial r_{c}

, as if the equations of motion came from a scalar WKB ansatz: there is no geometrical effect for these wave rays; that is to say, one scalar field—for instance, the geostrophic streamfunction—contains all the information on the phase of the fields. Therefore, first-order corrections to the Rossby wave ray dynamics only involve a corrected dynamical phase.

As for the higher-frequency Poincaré wave bands n = ±1, we can straightforwardly infer the properties of the one from the other: since the fields are real, a wave at +k on band +1 must behave as a wave at −k on band −1, which corresponds to the particle-hole-like symmetry (ω, k) → ( − ω, − k), because a real inertia-gravity wave packet is a combination of the two.⁴ Keeping that in mind we will work with n = +1 only. Expressions of F^(±1) and Ω^(±1) are, respectively, given in appendices A and B.

All terms combined, equations (3.13) finally add up to

\begin{aligned} {\dot{x}}_{c} = \frac{k_{x c}}{\sqrt{f {(y_{c})}^{2} + k_{c}^{2}}} + \frac{β (y_{c})}{2 (f {(y_{c})}^{2} + k_{c}^{2})}, \end{aligned}

4.3a

\begin{aligned} {\dot{y}}_{c} = \frac{k_{y c}}{\sqrt{f {(y_{c})}^{2} + k_{c}^{2}}}, \end{aligned}

4.3b

\begin{aligned} {\dot{k}}_{x c} = 0 \end{aligned}

4.3c

\begin{aligned} and & {\dot{k}}_{y c} = - \frac{β (y_{c}) f (y_{c})}{\sqrt{f {(y_{c})}^{2} + k_{c}^{2}}} . \end{aligned}

4.3d

Equation (4.3c) is a straightforward consequence of the invariance along the zonal direction whereas there is an effective force along the meridional direction. The only footprint of the medium’s variation on the group velocity of inertia-gravity wave rays is along the zonal direction, while the geometrical corrections compensate for the dynamical one for

{\dot{y}}_{c}

. On a clockwise rotating planet, up to the definition of the north pole, β is positive everywhere and vanishes at the poles. Therefore, only an eastward anomalous velocity remains along x and a pseudo-force

{\dot{k}}_{c}

along y in ray-tracing equations (4.3), both of which are proportional to β and thus of order ε in amplitude. There is an interesting parallel between this anomalous velocity under the β-effect and the anomalous Hall effect [6,35]: in this analogy, the packet undergoes an electric field proportional to β along the meridional direction y. As a consequence, an additional ‘anomalous’ velocity arises, proportional and perpendicular to the electric field.

(c) Comparison with scalar ray tracing

As discussed in the Introduction, the method used in [23,27–30] to derive ray-tracing equations for v does not deal with the geometrical aspects of the problem as it ignores the polarization relations. We shall recall here the results obtained from this scalar framework and compare them with those that we obtained with the vectorial ansatz and the variational principle used in §3.

Following Bretherton & Garrett [26], shallow-water ray-tracing equations in the presence of a varying Coriolis parameter are derived from the f-plane dispersion relation ω_n(r, k) given by expressions (2.10),

\dot{r} = \frac{\partial ω_{n}}{\partial k} and \dot{k} = - \frac{\partial ω_{n}}{\partial r} .

4.4

Ray coordinates are noted from now on as r = (x, y) and k = (k_x, k_y) as well, without the index c, for simplification. Equations (4.4) yield a term of order β for

\dot{k}

but no such correction in the expression of the group velocity

\dot{r}

. In order to compute the β correction to the shallow-water waves’ group velocity, the traditional approach [23,27–30] consists in applying Bretherton’s method to the exact scalar equation for v that includes the β-effect,

\partial_{t t t} v + f {(y)}^{2} \partial_{t} v - Δ \partial_{t} v - β (y) \partial_{x} v = 0.

4.5

The corrected dispersion relation is then obtained by formally replacing ∂_t with −iω and

\nabla = (\partial_{x}, \partial_{y})

with ik = (ik_x, ik_y) in (4.5), which yields

ω^{3} - (f {(y)}^{2} + k_{x}^{2} + k_{y}^{2}) ω - β (y) k_{x} = 0.

4.6

By differentiating equation (4.6) with respect to k, one eventually gets the following expression for the group velocity:

\frac{\partial ω}{\partial k} = \frac{k + (β / 2 ω) {\hat{e}}_{x}}{ω + (β k_{x} / 2 ω^{2})},

4.7

where

{\hat{e}}_{x}

is the unitary basis vector along the local zonal direction, pointing eastward, and ω is one of the solutions of the dispersion relation (4.6). This result can be found in Vallis’s book [23, p. 311]. It echoes directly the scalar WKB solution of equation (4.5). However, using only this equation for ray tracing is problematic whenever β ≠ 0, because the other fields of the problem satisfy different equations,

\partial_{t t t} u + f {(y)}^{2} \partial_{t} u - Δ \partial_{t} u + β (y) \partial_{x} u = - 2 β (y) \partial_{y} v - β^{'} (y) v

4.8a

and

\partial_{t t t} η + f {(y)}^{2} \partial_{t} η - Δ \partial_{t} η + β (y) \partial_{x} η = 2 β (y) f (y) v .

4.8b

This multi-component aspect leads to inconsistent WKB ansatzes for the different scalar fields of the problem [4]. Yet accounting for the three fields is crucial for computing ray observables such as the ones defined by relations (3.3) or (3.4), and therefrom to properly derive ray-tracing equations.

When β ∼ ε, both scalar and vectorial methods yield the same leading-order ray-tracing equation through the f-plane dispersion relation, whose solutions are given by expressions (2.10). The β term on the l.h.s. of equation (4.5) appears only at next order. We explained that β also induces additional geometric corrections to ray-tracing equations, owing to the multi-component nature of the problem. The scalar approach is therefore inconsistent at order ε.

Thus the only way to properly take into account the inhomogeneity parameter β is to introduce it as a perturbation in the vectorial problem. Keeping that in mind, it is instructive to compare the group velocity (4.7) derived within the traditional scalar WKB framework with the group velocity (4.3) derived within our vectorial framework. By differentiating expression (4.6) with respect to k and r and expanding the result at order 1 in β ∼ ε, one gets the following ray-tracing equations for Poincaré wave rays:⁵

\dot{r} = \frac{\partial ω}{\partial k} = \frac{k}{\sqrt{f^{2} + k^{2}}} + β [\frac{{\hat{e}}_{x}}{2 (f^{2} + k^{2})} - \frac{k_{x} k}{{(f^{2} + k^{2})}^{2}}]

4.9a

and

\dot{k} = - \frac{\partial ω}{\partial r} = - β \frac{f {\hat{e}}_{y}}{\sqrt{f^{2} + k^{2}}} .

4.9b

The latter must be compared with expressions (4.3a) and (4.3b) obtained with the vectorial ansatz. The most striking difference is that the correction proportional to β in the group velocity (4.9a) is not always pointing eastward, contrary to what is predicted by the vectorial method. Accounting for the geometrical character of the inertia-gravity wave rays thus adds a systematic eastward term to its group velocity compared with the expected value of the scalar WKB approximation, as illustrated in figure 4. Figure 4 presents the case of equatorial oscillation [23,28,30], an instructive example showing the enhanced eastward group velocity of an inertia-gravity wave packet, predicted by accounting for the Berry curvature of the Poincaré band in ray-tracing equations. However, one must keep in mind that in the equatorial region—as opposed to mid-latitude ray tracing (figure 3)—below a typical latitude called the equatorial radius of deformation (

L_{eq} = \sqrt{c / β_{eq}}

, for the first baroclinic mode in the equatorial ocean on Earth we have c ≃ 2 ms⁻¹, thus L_eq ≃ 300 km [23]), our scaling fails. On the one hand the Coriolis parameter vanishes at the equator, thus so does the separation in frequency between inertia-gravity modes and planetary modes, which can hybridize. On the other hand the wave packet is subjected to strong dispersion in the equatorial region, which eventually spreads its envelope to a scale comparable to L.

Figure 4. Ray trajectory of an inertia-gravity wave packet oscillating eastward around the equator, predicted by the different theories (same colour code as in figure 3). The initial wavelength is 2.8, β = 0.25 and the ray trajectories are plotted to t = 240. The vectorial theory anticipates a packet heading east to be faster than one within the scalar framework. Conversely, a packet heading west would be slower. (Online version in colour.)

One can wonder what the typical orders of magnitude are for this vectorial correction in dimensional units, say for oceanic inertia-gravity waves on Earth, with c = 2 ms⁻¹ and β = 2.3 × 10⁻¹¹ m⁻¹s⁻¹ (in the following all quantities are given in their natural units). Subtracting the projection along x of equation (4.9a) to expression (4.3a), the zonal projection of the difference in group velocity between the vectorial and the scalar method reads

Δ v_{x} = \frac{β k_{x}^{2}}{{({(f / c)}^{2} + k^{2})}^{2}} \sim β λ^{2} \sim c {(\frac{λ}{L_{eq}})}^{2},

4.10

which is a constant of motion with both methods, and shall be compared with the leading-order zonal velocity

c k_{x} / \sqrt{{(f / c)}^{2} + k^{2}}

. For inertia-gravity waves of wavelength smaller than the equatorial radius of deformation (λ ∼ 100 km), relation (4.10) adds up to a vectorial correction of about 1–10% of c, typically, to the zonal group velocity, thus yielding an overall zonal shift of order λ after a time t = λ/Δv_x ∼ (βλ)⁻¹, which is typically a month. The corresponding total zonal displacement of the wave packet after such a time is about c/(βλ) ∼ 6000 km.

5. Conclusion

Using the classical example of the linear β-plane shallow-water model, we showed how the vectorial nature of eigenmodes is involved in their ray dynamics, through both the β-effect and a geometrical observable: the Berry curvature. In a way that could not be derived from a scalar approach, these joint contributions yield an eastward deviation of the Poincaré wave rays’ trajectory, exhibiting a geophysical analogue to the anomalous Hall effect. The ray-tracing equations that we derive are based on coherent hypotheses with respect to the scales characterizing the wave packet and the variations of the Coriolis parameter, in contrast with scalar derivations of equatorial ray-tracing equations that can be found in classical textbooks.

Previous studies have revealed geometric phase effects in geophysical waves [3,4,34], although they are rarely mentioned in geophysical fluid dynamics textbooks, with the notable exception of [25]. Our contribution here has been to exhibit a quantitative manifestation of the Berry curvature in ray-tracing equations, with application to shallow-water waves. This manifestation is subtle from a practical point of view. Indeed, the Berry curvature of inertia-gravity wave bands is highest at small wave vectors, and thus for large wavelength (figure 2). Yet our analysis is based on a scaling that assumes sufficiently small wavelength. Furthermore, a large Berry curvature is concomitant with large dispersion as both effects are associated with the proximity of a degeneracy point between wave bands. In practice, large dispersion tends to spread the envelop of the wavepacket. This effect of dispersion, together with dissipation and nonlinearities, will need to be addressed in future works.

Our approach thus highlights the multi-component nature of geophysical wave dynamics, where the primitive set of equations always involve the time derivative of several fields coupled together. The dynamics is sometimes reduced down to a scalar equation, which can conveniently be expressed in a Lagrangian framework different from the one used in this paper [39]. These scalar field Lagrangian approaches are very useful as they allow one to describe nonlinear effects. However, information on the underlying multi-component dynamics is not encoded into such scalar Lagrangians. This is why we used here an alternative variational approach that takes into account the multi-component nature of the wave equations. It cannot account for nonlinear effects, but it is suitable for capturing Berry curvature effects on ray trajectories.

Since the Berry curvature most often comes along with multi-component wave problems in the presence of broken discrete symmetries, it is natural to expect its manifestation on a wider class of systems, from small-scale hydrodynamics to large-scale geophysics and astrophysics, as well as in the domain of seismic and elastic waves.

Data accessibility

We declare that all information required to reproduce the paper is contained within the paper.

Authors' contributions

N.P. conducted the research and performed the numerical calculations. P.D. and A.V. supervised the study. All authors discussed the physics, participated in writing and revising the manuscript, gave final approval for publication and agree to be held accountable for the work performed herein.

Competing interests

We declare we have no competing interests.

Funding

This project was supported by national grant no. ANR-18-CE30-0002-01. N.P. was funded by a PhD grant allocation Contrat doctoral Normalien.

Appendix A. Eigenmodes and Berry curvature of the f-plane shallow-water system

We give here an expression of the eigenmodes and Berry curvature of the f-plane shallow-water model for both geostrophic (n = 0) and inertia-gravity (n = ±1) wave bands, which are purely geometrical features of the eigenspaces supporting the vectorial plane-wave solutions of the f-plane problem.

As shown in §2., looking for dispersion and polarization relations of the Fourier solutions of the f-plane shallow-water model amounts to solving the eigenvalue equation (2.9). For a given value of (f, k_x, k_y), the eigenvalues are given by expressions (2.10), and the corresponding $C^{3}$ basis of normalized eigenmodes is

U_{0} [f, k_{x}, k_{y}] = \frac{1}{\sqrt{f^{2} + k^{2}}} (\begin{matrix} k_{y} \\ - k_{x} \\ i f \end{matrix}) = (\begin{matrix} \sin θ \sin ϕ \\ - \sin θ \cos ϕ \\ i \cos θ \end{matrix}),

A 1

for the geostrophic band, and

U_{\pm 1} [f, k_{x}, k_{y}] = \frac{1}{k \sqrt{2}} (\begin{matrix} k_{x} \pm i \frac{f k_{y}}{\sqrt{f^{2} + k^{2}}} \\ k_{y} \mp i \frac{f k_{x}}{\sqrt{f^{2} + k^{2}}} \\ \pm \frac{k^{2}}{\sqrt{f^{2} + k^{2}}} \end{matrix}) = \frac{1}{\sqrt{2}} (\begin{matrix} \cos ϕ \pm i \cos θ \sin ϕ \\ \sin ϕ \mp i \cos θ \cos ϕ \\ \pm \sin θ \end{matrix}),

A 2

for inertia-gravity bands, all three being defined up to a gauge phase. Owing to the linear property H[f, k_x, k_y] = −H[ − f, − k_x, − k_y] we have U₋₁[f, k_x, k_y] = −U₊₁[ − f, − k_x, − k_y], again up to a choice of gauge. Above we defined the norm

k \equiv | | k | | = \sqrt{k_{x}^{2} + k_{y}^{2}}

and the spherical coordinates (ϕ, θ) of the

R^{3}

vector

{(\begin{matrix} k_{x} k_{y} f \end{matrix})}^{t}

of norm

\sqrt{f^{2} + k^{2}}

. From expressions (2), we arrive at

i U_{0}^{†} d U_{0} = 0, i U_{\pm 1}^{†} d U_{\pm 1} = \mp \cos θ d ϕ,

A 3

for the connections of the eigenbundles, hence

F^{(0)} = 0, F^{(\pm 1)} = \pm \sin θ d θ \land d ϕ,

A 4

for the Berry curvatures. Keeping in mind that in the three-dimensional case the coefficients of

F^{(n)}

are given by the curl of

i U_{n}^{†} \nabla U_{n}

, which is equal to

\mp (\cot θ / \sqrt{f^{2} + k^{2}}) {\hat{e}}_{ϕ}

if n = ±1, we have for the latter

(\begin{matrix} F_{k_{y} f}^{(\pm 1)} \\ F_{f k_{x}}^{(\pm 1)} \\ F_{k_{x} k_{y}}^{(\pm 1)} \end{matrix}) = \nabla \times (i U_{\pm 1}^{†} \nabla U_{\pm 1}) = \pm \frac{1}{{\sqrt{f^{2} + k^{2}}}^{3}} (\begin{matrix} k_{x} \\ k_{y} \\ f \end{matrix}),

A 5

in the natural Cartesian basis of

{(\begin{matrix} k_{x} k_{y} f \end{matrix})}^{t}

. Finally, formally replacing the variable f in expression (5), owing to df = β dy, yields the expressions of the only non-zero coefficients⁶ of the Berry curvature F^(±1), used to derive ray-tracing equations (4.3)

\begin{aligned} F_{k_{y} y}^{(\pm 1)} (r, k) = - F_{y k_{y}}^{(\pm 1)} (r, k) = \pm \frac{β k_{x}}{{\sqrt{f {(y)}^{2} + k^{2}}}^{3}}, \end{aligned}

A 6a

\begin{aligned} F_{y k_{x}}^{(\pm 1)} (r, k) = - F_{k_{x} y}^{(\pm 1)} (r, k) = \pm \frac{β k_{y}}{{\sqrt{f {(y)}^{2} + k^{2}}}^{3}} \end{aligned}

A 6b

\begin{aligned} and & F_{k_{x} k_{y}}^{(\pm 1)} (r, k) = - F_{k_{y} k_{x}}^{(\pm 1)} (r, k) = \pm \frac{f (y)}{{\sqrt{f {(y)}^{2} + k^{2}}}^{3}} . \end{aligned}

A 6c

Appendix B. Variational derivation of ray-tracing equations

In this appendix, we derive the ray equations for ray tracing starting from ansatz (3.1) for the wave packet. We pay particular attention to the necessary approximations to get the expansion of the Lagrangian (3.7).

(a) Reduction of the Lagrangian $L^{'}$

Firstly, definition (3.6) can be separated as

L^{'} = \frac{⟨ ψ^{'} | i \partial_{t} | ψ^{'} ⟩}{⟨ ψ^{'} | ψ^{'} ⟩} - \frac{⟨ ψ^{'} | \hat{H} | ψ^{'} ⟩}{⟨ ψ^{'} | ψ^{'} ⟩} .

B 1

The second term, defined in relation (3.8), will be treated separately. Using the ansatz (3.1), the first term yields

\frac{⟨ ψ^{'} | i \partial_{t} | ψ^{'} ⟩}{⟨ ψ^{'} | ψ^{'} ⟩} = \frac{\int A^{2} (- \partial_{t} Φ + i U_{n}^{†} \partial_{t} U_{n})}{\int A^{2}} + \frac{i}{2} \frac{d}{d t} \ln (\int A^{2}),

B 2

with the notation

\int A^{2} \equiv \int d^{2} r A {(r, t)}^{2}

. To simplify the presentation, variables of functions have been omitted, i.e. Φ stands for Φ(r, t) and U_n for U_n[f(r), k_c(t)], for instance. Since the operator

\hat{H}

is Hermitian, the norm of |ψ〉 is left constant by the dynamics;⁷ therefore, the second term in equation (8) is null, leaving us only with the first term, which is real. Similarly, the expression of the mean wave vector defined in equations (3.4) is simplified into

k_{c} (t) = \frac{\int A^{2} (\nabla Φ - i U_{n}^{†} \nabla U_{n})}{\int A^{2}} .

B 3

Let us now simplify expressions (8) and (9) by approximating the integrals weighted by A². The Taylor expansion of a general function g(r, t) around r_c(t) reads

g (r, t) = g (r_{c}, t) + (r - r_{c}) \cdot \nabla g (r_{c}, t) + O ({(r - r_{c})}^{2} ε^{- 2} g_{0}),

B 4

if ε⁻¹ is the variation scale of the function g of order g₀. By performing the integral of equation (10) weighted by A², the second term on the r.h.s. of equation (10) yields 0, by definition of r_c, whereas by definition of α the third one remains and we get

\frac{\int A^{2} g (r, t)}{\int A^{2}} = g (r_{c}, t) + O ({(\frac{ε}{α})}^{2} g_{0}) .

B 5

The derivatives of the slowly varying functions in integrals (8) and (9) reveal terms of order 1, with −∂_tΦ₀/ε = −∂_TΦ₀ and

\nabla Φ_{0} / ε = \nabla_{R} Φ_{0}

, and ε with

ε (- \partial_{T} Φ_{1} + i U_{n}^{†} \partial_{T} U_{n})

and

ε (\nabla_{R} Φ_{1} - i U_{n}^{†} \nabla_{R} U_{n})

. Therefore, we have here g₀ of order 1 at most and we arrive at

\frac{⟨ ψ^{'} | i \partial_{t} | ψ^{'} ⟩}{⟨ ψ^{'} | ψ^{'} ⟩} = - \partial_{t} Φ (r_{c}, t) + i U_{n} {[f (y_{c}), k_{c} (t)]}^{†} \partial_{t} U_{n} [f (y_{c}), k_{c} (t)] + O ({(\frac{ε}{α})}^{2})

B 6

and

k_{c} = \nabla Φ (r_{c}, t) - i U_{n} {[f (y_{c}), k_{c} (t)]}^{†} \nabla U_{n} [f (y_{c}), k_{c} (t)] + O ({(\frac{ε}{α})}^{2}) .

B 7

This last expression is the same as (3.5), only here for the purposes of the following development we do not need to separate the terms of order 1 and ε. Noticing now that (d/dt)ϕ(r_c(t), t) is equal to

(\partial_{t} + {\dot{r}}_{c} \cdot \nabla) ϕ (r_{c}, t)

for any function ϕ(r, t), we can combine equation (12) with equation (13) and expressions (3.8), and the Lagrangian reads

L^{'} + \frac{d}{d t} Φ (r_{c} (t), t) = {\dot{r}}_{c} \cdot k_{c} - Ω^{(n)} (r_{c}, k_{c}) + i U_{n} {(r_{c}, k_{c})}^{†} \frac{d}{d t} U_{n} (r_{c}, k_{c}) + O ({(\frac{ε}{α})}^{2}) .

B 8

This last relation corresponds to expression (3.7) with the notation χ(t) ≡ Φ(r_c(t), t).

(b) Expression of the ray Hamiltonian

Let us now prove expression (3.15) for Ω⁽ⁿ⁾ in the case of the shallow-water model. First of all, we note that the Hamiltonian operator defined in relations (2.8) reads $\hat{H} (r, \nabla) = f (r) (\partial H / \partial f) - i (\partial H / \partial k) \cdot \nabla$ , using the formal derivatives of the symbol H defined in relation (2.9). In addition, one can check that $U_{n}^{†} (\partial H / \partial k) U_{n}$ is equal to c_n ≡ ∂ω_n/∂k, the group velocity of the nth band of the f-plane system. Then using ansatz (3.1) yields

\begin{aligned} ⟨ \hat{H} ⟩ & \equiv \frac{⟨ ψ | \hat{H} | ψ ⟩}{⟨ ψ | ψ ⟩} \\ = \frac{\int A^{2} [c_{n} \cdot \nabla Φ + f U_{n}^{†} (\partial H / \partial f) U_{n} - i U_{n}^{†} (\partial H / \partial k) \cdot \nabla U_{n}] - i c_{n} \cdot \nabla (A^{2} / 2)}{\int A^{2}} . \end{aligned}

B 9

Integrating by parts the last term in the integral of expression (15), and noticing that the symbol reads

H [f (r), k_{c}] = f (r) (\partial H / \partial f) + \frac{\partial H}{\partial k} \cdot k_{c}

, equation (15) yields

⟨ \hat{H} ⟩ = \frac{\int A^{2} [ω_{n} + c_{n} \cdot (\nabla Φ - k_{c}) + i / 2 \nabla \cdot c_{n} - i U_{n}^{†} (\partial H / \partial k) \cdot \nabla U_{n}]}{\int A^{2}} .

B 10

In addition, knowing that

\sum_{m \in {- 1, 0, + 1}} U_{m} U_{m}^{†} = I_{3}

(the 3-by-3 identity matrix) and that the connection

i U_{n}^{†} \nabla U_{n}

is real, one gets

\frac{i}{2} \nabla \cdot c_{n} - i U_{n}^{†} \frac{\partial H}{\partial k} \cdot \nabla U_{n} = - c_{n} \cdot i U_{n}^{†} \nabla U_{n} + I m (\sum_{m \neq n} U_{n}^{†} \frac{\partial H}{\partial k} U_{m} \cdot U_{m}^{†} \nabla U_{n}) .

B 11

For m ≠ n, one can prove that

U_{m}^{†} \nabla U_{n}

is equal to

U_{m}^{†} \nabla H U_{n} / (ω_{n} - ω_{m})

[2]; therefore, relation (16) yields

\begin{aligned} ⟨ \hat{H} ⟩ = (\begin{matrix} \int A^{2} [ω_{n} + c_{n} \cdot (\nabla Φ - k_{c} - i U_{n}^{†} \nabla U_{n}) \\ - I m [\sum_{m \neq n} (U_{n}^{†} (\partial H / \partial r) U_{m} \cdot U_{m}^{†} (\partial H / \partial k) U_{n} / (ω_{n} - ω_{m})]] \end{matrix}) / \int A^{2} . \end{aligned}

B 12

With the approximation of integrals weighted by A² explained in appendix Aa, the second term in expression (18) being straightforward to deal with via relation (13) and the third one of order ε, equation (18) yields

\begin{aligned} ⟨ \hat{H} ⟩ & = ω_{n} - I m (\sum_{m \neq n} \frac{U_{n}^{†} (\partial H / \partial r_{c}) U_{m} \cdot U_{m}^{†} (\partial H / \partial k_{c}) U_{n}}{ω_{n} - ω_{m}}) + O ({(\frac{ε}{α})}^{2}) \\ \equiv Ω^{(n)} + O ({(\frac{ε}{α})}^{2}) . \end{aligned}

B 13

An equivalent writing of Ω⁽ⁿ⁾, as in [14], is given by

Ω^{(n)} = ω_{n} - I m [\frac{\partial U_{n}^{†}}{\partial r_{μ c}} (ω_{n} I_{3} - H) \frac{\partial U_{n}}{\partial k_{μ c}}] .

B 14

All functions in expressions (20) are valued at (r_c, k_c), which demonstrates that Ω⁽ⁿ⁾ is indeed a function of r_c and k_c only, at leading order in the gradient corrections. This proves relation (3.15). Using expression (19) for Rossby waves (n = 0) one gets relation (4.2) for Ω. For Poincaré waves (n = ±1), it yields

Ω^{(\pm 1)} (r_{c}, k_{c}) = \pm \sqrt{f {(y_{c})}^{2} + k_{c}^{2}} - \frac{β k_{x c}}{2 (f {(y_{c})}^{2} + k_{c}^{2})} .

B 15

Footnotes

1 Indeed r_c and k_c are both seen as generalized coordinates in phase space, but in the particular case of a null connection we have the equality $k_{c} = \partial L / \partial {\dot{r}}_{c}$ ; therefore, k_c identifies to the conjugated momentum of r_c.

2 This corresponds to the ray Hamiltonian, which has the dimension of a frequency and is not to be confused with the energy Hamiltonian commonly used in variational methods for fluid dynamics [26,38,39].

3 The same perturbative expressions for equations (3.14) and (3.15) can be found in [2,14] for corrected Bloch bands, owing to the interaction between an external magnetic field and the orbital magnetic moment of an electron, for instance.

4 This can be proven by injecting relation U₋₁[f, k_x, k_y] = −U₊₁[ − f, − k_x, − k_y] (see appendix A) into expressions (3.14) and (3.15), and then the latter into ray-tracing equations (3.13).

5 Again we work with the positive solution (n = +1), which reads $ω (r, k) = \sqrt{f {(y)}^{2} + k^{2}} + (β k_{x} / 2 (f {(y)}^{2} + k^{2})) + O (ε^{2})$ .

6 Since the Coriolis parameter has no dependence in the longitudinal direction x and F⁽ⁿ⁾ are antisymmetric tensors, the Berry curvature is entirely defined by the three coefficients $F_{y k_{x}}^{(\pm 1)}$ , $F_{y k_{y}}^{(\pm 1)}$ and $F_{k_{x} k_{y}}^{(\pm 1)}$ .

7 Indeed, $(i / 2) (d / d t) \ln (\int A^{2}) = i (d / d t) | | ψ^{'} | |$ and we can restrain the search for a solution to the space of functions of constant norm with respect to time, because $\hat{H}$ is Hermitian: in that way the Lagrangian $L^{'}$ is real. One would directly remove the second term in equation (8) from the variational problem because it is an exact time derivative, but if the Lagrangian were not real the Schrödinger-like equation (2.7) would not be equivalent to the corresponding variational problem [19].

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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences cover image

April 2021

Volume 477Issue 2248

Article Information

DOI:https://doi.org/10.1098/rspa.2020.0844
Published by:Royal Society
Online ISSN:1471-2946

History:

Manuscript received19/10/2020
Manuscript accepted29/03/2021
Published online21/04/2021
Published in print28/04/2021

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Manifestation of the Berry curvature in geophysical ray tracing

Abstract

1. Introduction

2. Formulation of the problem for shallow-water waves

(a) The linearized rotating shallow-water model

(b) Dimensionless parameters of the problem and slow variables

(c) Spectral and geometrical features of waves in the f-plane model

3. Berry curvature in ray tracing of multi-component waves

(a) A ray-tracing method based on a variational principle for multi-component wave problems

(b) Ray-tracing equations and the Berry curvature

4. Application to shallow-water wave packets

(a) β-plane shallow-water waves

(b) Berry curvature contribution to eastward drift of inertia-gravity waves

(c) Comparison with scalar ray tracing

5. Conclusion

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Authors' contributions

Competing interests

Funding

Appendix A. Eigenmodes and Berry curvature of the f-plane shallow-water system

Appendix B. Variational derivation of ray-tracing equations

Footnotes

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