Three-dimensional geophysical fluids support both internal and boundary-trapped waves. To obtain the normal modes in such fluids, we must solve a differential eigenvalue problem for the vertical structure (for simplicity, we only consider horizontally periodic domains). If the boundaries are dynamically inert (e.g., rigid boundaries in the Boussinesq internal wave problem and flat boundaries in the quasigeostrophic Rossby wave problem), the resulting eigenvalue problem typically has a Sturm–Liouville form and the properties of such problems are well-known. However, when restoring forces are also present at the boundaries, then the equations of motion contain a time-derivative in the boundary conditions, and this leads to an eigenvalue problem where the eigenvalue correspondingly appears in the boundary conditions. In certain cases, the eigenvalue problem can be formulated as an eigenvalue problem in the Hilbert space L2C and this theory is well-developed. Less explored is the case when the eigenvalue problem takes place in a Pontryagin space, as in the Rossby wave problem over sloping topography. This article develops the theory of such problems and explores the properties of wave problems with dynamically active boundaries. The theory allows us to solve the initial value problem for quasigeostrophic Rossby waves in a region with sloping bottom (we also apply the theory to two Boussinesq problems with a free-surface). For a step-function perturbation at a dynamically active boundary, we find that the resulting time-evolution consists of waves present in proportion to their projection onto the dynamically active boundary.

An important tool in the study of wave motion near a stable equilibrium is the separation of variables. When applicable, this elementary technique transforms a linear partial differential equation into an ordinary differential eigenvalue problem for each coordinate [e.g., Hillen et al. (2012)]. Upon solving the differential eigenvalue problems, one obtains the normal modes of the physical system. The normal modes are the fundamental wave motions for the given restoring forces, each mode represents an independent degree of freedom in which the physical system can oscillate, and any solution of the wave problem may be written as a linear combination of these normal modes.

To derive the normal modes, we must first linearize the dynamical equations of motion about some equilibrium state. We then encounter linearized restoring forces of two kinds as follows:

  1. volume-permeating forces experienced by fluid particles in the interior, and

  2. boundary-confined forces only experienced by fluid particles at the boundary.

Examples of volume-permeating forces include the restoring forces resulting from continuous density stratification and continuous volume potential vorticity gradients. These restoring forces, respectively, result in internal gravity waves (Sutherland, 2010) and Rossby waves (Vallis, 2017). Examples of boundary-confined restoring forces include the gravitational force at a free-surface (i.e., at a jump discontinuity in the background density), forces arising from gradients in surface potential vorticity (Schneider et al., 2003), and the molecular forces giving rise to surface tension. These restoring forces, respectively, result in surface gravity waves (Sutherland, 2010), topographic/thermal waves (Hoskins et al., 1985), and capillary waves (Lamb, 1975).

In the absence of boundary-confined restoring forces, we can often apply Sturm–Liouville theory [e.g., Hillen et al. (2012); Zettl (2010)] to the resulting eigenvalue problem. We thus obtain a countable infinity of waves whose vertical structures form a basis of L2, the space of square-integrable functions (see Sec. II), and given some initial vertical structure, we know how to solve for the subsequent time-evolution as a linear combination for linearly independent waves. Moreover, a classic result of Sturm–Liouville theory is that the nth mode has n internal zeros.

In the presence of boundary-confined restoring forces, the governing equations have a time-derivative in the boundary conditions. The resulting eigenvalue problem correspondingly contains the eigenvalue parameter in the boundary conditions. Sturm–Liouville theory is inapplicable to such problems.

In this article, we present a general method for solving these problems by delineating a generalization of Sturm–Liouville theory. Some consequences of this theory are the following. There is a countable infinity of waves whose vertical structures form a basis of L2Cs, where s is the number of dynamically active boundaries; thus, each boundary-trapped wave, in mathematically rigorous sense, provides an additional degree of freedom to the problem. The modes satisfy an orthogonality relation involving boundary terms, the modes may have a negative norm, and the modes may have finite jump discontinuities at dynamically active boundaries (although the solutions are always continuous, see Sec. III C). When negative norms are possible (as in quasigeostrophic theory), there is a new expression for the Fourier coefficients that one must use to solve initial value problems [see Eq. (23)]. We can also expand boundary step-functions (representing some boundary localized perturbation) as a sum of modes. Moreover, the nth mode may not have n internal zeros; indeed, depending on physical parameters in the problem, two or three linearly independent modes with an identical number of internal zeros may be present.

We also show that the eigenfunction expansion of a function is term-by-term differentiable, with the derivative series converging uniformly on the whole interval, regardless of the boundary condition the function satisfies at the dynamically active boundaries. This property is in contrast with a traditional Sturm–Liouville eigenfunction expansion where the term-by-term derivative converges uniformly only if the function satisfies the same boundary condition as the eigenfunctions.

We apply the theory to three geophysical wave problems. The first is that of a Boussinesq fluid with a free-surface; we find that the nth mode has n internal zeros. The second example is that of a rotating Boussinesq fluid with a free-surface where we assume that the stratification suppresses rotational effects in the interior but not at the upper boundary. We find that there are two linearly independent modes with M internal zeros, where the integer M depends on the ratio of the Coriolis parameter to the horizontal wavenumber, and that the eigenfunctions have a finite jump discontinuity at the upper boundary. The third application is to a quasigeostrophic fluid with a sloping lower boundary. We find that modes with an eastward phase speed have a negative norm, whereas modes with a westward phase speed have a positive norm (the sign of the norm has implications for the relative phase of a wave and for series expansions). Moreover, depending on the propagation direction, there can be two linearly independent modes with no internal zeros. For all three examples, we outline the properties of the resulting series expansions and provide the general solution. We also consider the time-evolution resulting from a vertically localized perturbation at a dynamically active boundary; we idealize such a perturbation as a boundary step-function. The step-function perturbation induces a time-evolution in which the amplitude of each constituent wave is proportional to the projection of that wave onto the boundary.

To our knowledge, most of the above results cannot be found in the literature [however, the gravity wave orthogonality relation has been noted before, e.g., Gill (1982); Kelly (2016) for the hydrostatic case and Olbers (1986); Early et al. (2020) for the non-hydrostatic case]. For instance, we provide the only solution to the initial value problem for Rossby waves over topography in the literature [Eq. (78)]. Moreover, many of the properties we discuss arise in practical problems in physical oceanography. The number of internal zeros of Rossby waves is also a useful quantity in observational physical oceanography [e.g., Clément et al. (2014); de La Lama et al. (2016)]. In addition, the question of whether the quasigeostrophic baroclinic modes are complete is a controversial one. Lapeyre (2009) has suggested that the baroclinic modes are incomplete because they assume a vanishing surface buoyancy anomaly. Consequently, Smith and Vanneste (2012) addressed this issue by deriving an L2C2 basis for quasigeostrophic theory. Yet, many authors, citing completeness theorems from Sturm–Liouville theory, insist that the baroclinic modes are indeed complete and can represent all quasigeostrophic states (Ferrari and Wunsch, 2010; LaCasce, 2012; and Rocha et al., 2015). This article shows that by including boundary-confined restoring forces, we obtain a set of modes with additional degrees-of-freedom. These degrees-of-freedom manifest in the behavior of eigenfunction expansions at the boundaries. In addition, the distinction between L2 and L2Cs bases that we present here is useful for equilibrium statistical mechanical calculations where one must decompose fluid motion onto a complete set of modes (Bouchet and Venaille, 2012; Venaille et al., 2012).

This article is organized as follows: We formulate the mathematical theory in Sec. II. We then apply the theory to the two Boussinesq wave problems (Sec. III) and the quasigeostrophic wave problem (Sec. IV). We consider the time-evolution of a localized perturbation at a dynamically active boundary in Sec. V. We then conclude in Sec. VI.

In this section, we outline the theory of the differential eigenvalue problem

(pϕ)+qϕ=λrϕforzz1,z2,
(1)
a1ϕ(z1)b1(pϕ)(z1)=λc1ϕ(z1)d1(pϕ)(z1),
(2)
a2ϕ(z2)b2(pϕ)(z2)=λc2ϕ(z2)d2(pϕ)(z2),
(3)

where p−1, q, and r are real-valued integrable functions; where ai, bi, ci, and di are real numbers with i ∈ {1, 2}; and where λC is the eigenvalue parameter. We further assume that p > 0 and r > 0, that p and r are twice continuously differentiable, that q is continuous, and that (ai, bi) ≠ (0, 0) for i ∈ {1, 2}. The system of equations (1)–(3) is an eigenvalue problem for the eigenvalue λC and differs from a regular Sturm–Liouville problem in that λ appears in the boundary conditions [(2) and (3)]. That is, setting ci = di = 0 recovers the traditional Sturm–Liouville problem. The presence of λ as part of the boundary condition leads to some fundamentally new mathematical features that are the subject of this section and fundamental to the physics of this study.

It is useful to define the following two boundary parameters:

Di=(1)i+1aidibici,i=1,2.
(4)

Just as the function r acts as a weight for the interval (z1, z2) in traditional Sturm–Liouville problems, the constants Di1 will play analogous roles for the boundaries z = zi when Di ≠ 0.

Outline of the mathematics. The right-definite case, when Di ≥ 0 for i ∈ {1, 2}, is well-known in the mathematics literature; most of the right-definite results in this section are due to Evans (1970); Walter (1973); and Fulton (1977). In contrast, the left-definite case, defined below, is much less studied. In this section, we generalize the right-definite results of Fulton (1977) to the left-definite problem as well as provide an intuitive formulation [in terms of functions rather than vectors, for a vector formulation, see Fulton (1977)] of the eigenvalue problem.

In Sec. II A, we state the conditions under which we obtain real eigenvalues and a basis of eigenfunctions. We proceed, in Sec. II B, to explore the properties of eigenfunctions and eigenfunction expansions. Finally, in Sec. II C, we discuss oscillation properties of the eigenfunctions. Additional properties of the eigenvalue problem are found in  Appendix A, and a literature review, along with various technical proofs, is found in  Appendix B.

1. The functions space of the problem

We denote by L2 the Hilbert space of square-integrable “functions” ϕ on the interval (z1, z2) satisfying

z1z2ϕ2rdz<.
(5)

To be more precise, the elements of L2 are not functions but rather equivalence classes of functions [e.g., Reed and Simon (1980), Sec. I C]. Two functions, ϕ and ψ, are equivalent in L2 (i.e., ϕ = ψ in L2) if they agree in a mean-square sense on [z1, z2],

z1z2ϕ(z)ψ(z)2rdz=0.
(6)

Significantly, we can have ϕ = ψ in L2 but ϕψ pointwise.

Furthermore, as a Hilbert space, L2 is endowed with a positive-definite inner product

ϕ,ψσ=z1z2ϕ*ψdσ=z1z2ϕ*ψrdz,
(7)

where the symbol * denotes complex conjugation and the measure σ associated L2 induces a differential element dσ = rdz (see  Appendix A). The positive-definiteness is ensured by our assumption that r > 0 (i.e., ϕ,ϕσ>0 for ϕ ≠ 0 when r > 0).

It is well-known that traditional Sturm–Liouville problems [i.e., Eqs. (1)–(3) with ci = di = 0 for i = 1, 2] are eigenvalue problems in some subspace of L2 (Debnath and Mikusinski, 2005). For the more general case of interest here, the eigenvalue problem occurs over a “larger” function space denoted by Lμ2, which we construct in  Appendix A.

Let the integer s ∈ {0, 1, 2} denote the number of λ-dependent boundary conditions, and let S denote the set

S={j|j{1,2} and (cj,dj)(0,0)}.
(8)

Here, S is one of ∅, {1}, {2}, {1, 2} and s is the number of elements in the set S. In  Appendix A, we show that Lμ2 is isomorphic to the space L2Cs and is thus “larger” than L2 by s dimensions.

We denote elements of Lμ2 by upper case letters Ψ; we define Ψ(z) for z ∈ [z1, z2] by

Ψ(z)=Ψ(zi)atz=zi,foriS,ψ(z)otherwise,
(9)

where Ψ(zi)C are constants for iS, and the corresponding lower case letter ψ denotes an element of L2. Two elements Φ and Ψ of Lμ2 are equivalent in Lμ2 if and only if

  1. Φ(zi) = Ψ(zi) for iS and

  2. ϕ(z) and ψ(z) are equivalent in L2 [i.e., as in Eq. (6)].

Here, Φ, as an element of Lμ2, is defined as in Eq. (9). The primary difference between L2 and Lμ2 is that Lμ2 discriminates between functions that disagree at λ-dependent boundaries.

The measure μ associated with Lμ2 (see  Appendix A) induces a differential element

dμ(z)=r(z)+iSDi1δ(zzi)dz,
(10)

where δ(z) is the Dirac delta. The induced inner product on Lμ2 is

Φ,Ψ=z1z2Φ*Ψdμ=z1z2Φ*Ψrdz+iSDi1Φ(zi)*Ψ(zi).
(11)

If Di > 0 for iS, then this inner product is positive-definite and Lμ2 is a Hilbert space. However, this is not the case in general.

Let κ denote the number of negative Di for iS (the possible values are κ = 0, 1, 2). Then, Lμ2 has a κ-dimensional subspace of elements Ψ satisfying

Ψ,Ψ<0.
(12)

This makes Lμ2 a Pontryagin space of index κ (Bognár, 1974). If κ = 0, then Lμ2 is again a Hilbert space. In the present case, Lμ2 also has an infinite-dimensional subspace of elements ψ satisfying

Ψ,Ψ>0.
(13)

2. Reality and completeness

In Appendix A 2, we reformulate the eigenvalue problem (1)–(3) as an eigenvalue problem of the form

LΦ=λΦ
(14)

in a subspace of Lμ2, where L is a linear operator and Φ is an element of Lμ2. We also define the notions of right- and left-definiteness that are required for the reality and completeness theorem. The following two propositions can be considered to define right- and left-definiteness for applications of the theory. Both propositions are obtained through straightforward manipulations (see  Appendix A).

Proposition II.1

(criterion for right-definiteness). The eigenvalue problem(1)–(3)is right-definite ifr > 0 andDi > 0 foriS.

Proposition II.2

(criterion for left-definiteness). The eigenvalue problem(1)–(3)is left-definite if the following conditions hold:

  • the functionsp, qsatisfyp > 0, q ≥ 0;

  • for theλ-dependent boundary conditions, we have
    aiciDi0,bidiDi0,(1)iaidiDi0foriS;
    (15)
  • for theλ-independent boundary conditions, we have
    bi=0or(1)i+1aibi0ifbi0fori{1,2}\S.
    (16)

The notions of right- and left-definiteness are not mutually exclusive. In particular, a problem can be neither right- nor left-definite, both right- and left-definite, only right-definite, or only left-definite. In this article, we always assume that p > 0 and r > 0.

The reality of the eigenvalues and the completeness of the eigenfunctions in the space Lμ2 is given by the following theorem:

Theorem II.3
(reality and completeness). Suppose the eigenvalue problem(1)–(3)is either right-definite or left-definite. Moreover, if the problem is not right-definite, we assume thatλ = 0 is not an eigenvalue. Then, the eigenvalue problem(1)–(3)has a countable infinity of real simple eigenvaluesλnsatisfying
λ0<λ1<<λn<,
(17)
with corresponding eigenfunctions Φn. Furthermore, the set of eigenfunctions{Φn}n=0is a complete orthonormal basis forLμ2satisfying
Φm,Φn=±δmn.
(18)

Proof.

See Appendix B 3.□

Recall that κ denotes the number of negative Di for iS. We then have the following corollary of the Proof of Theorem II.3:

Proposition II.4.
Suppose the eigenvalue problem(1)–(3)is left-definite and thatλ = 0 is not an eigenvalue. Then, there areκnegative eigenvalues and their eigenfunctions satisfy
Φ,Φ<0.
(19)
The remaining eigenvalues are positive and their eigenfunctions satisfy
Φ,Φ>0.
(20)

In other words, Proposition II.4 states that we have the relationship

λnΦn,Φn>0
(21)

for left-definite problems.

For the remainder of Sec. II, we assume that the eigenvalue problem (1)–(3) satisfies the requirements of Theorem II.3.

1. Eigenfunction expansions

The eigenvalue problem (1)–(3) has eigenfunctions{Φn}n=0 as well as corresponding solutions{ϕn}n=0. In other words, while the ϕn are the solutions to the differential equation defined by Eqs. (1)–(3) with λ = λn, the eigenfunctions required by the operator formulation of the problem [Eq. (14)] are Φn. The functions Φn and ϕn are related by Eq. (9), with the boundary values Φn(zi) of Φn determined by

Φn(zi)=ciϕ(z)di(pϕ)(z)foriS.
(22)

Thus, while the solutions ϕn are continuously differentiable over the closed interval [z1, z2], the eigenfunctions Φn are continuously differentiable over the open interval (z1, z2) but generally have finite jump discontinuities at the λ-dependent boundaries. The eigenfunctions Φn are continuous in the closed interval [z1, z2] only if ci = 1 and di = 0 for iS. In this case, the eigenfunctions Φn coincide with the solutions ϕn on the closed interval [z1, z2].

The boundary conditions of the eigenvalue problem (1)–(3) are not unique. One can multiply each boundary condition by an arbitrary constant to obtain an equivalent problem. To uniquely specify the eigenfunctions in physical applications, the boundary coefficients {ai, bi, ci, di} of Eqs. (1)–(3) must be chosen so that rdz has the same dimensions as Di1δ(zzi)dz [recall that δ(z) has the dimension of inverse length]. In the quasigeostrophic problem, we must also invoke continuity and set ci = 1.

Since {Φn}n=0 is a basis for Lμ2, then any ΨLμ2 may be expanded in terms of the eigenfunctions (Bognár, 1974, Theorem IV.3.4),

Ψ=n=0Ψ,ΦnΦn,ΦnΦn.
(23)

We emphasize that the above equality is an equality in Lμ2 and not a pointwise equality [see the discussion following Eq. (9)]. Some properties of Lμ2 expansions are given in Appendix A 3.

An important property that distinguishes the basis {Φn}n=0 of Lμ2 from an L2 basis is its “sensitivity” to function values at boundary points z = zi for iS. See Sec. V for a physical application.

A natural question is whether the basis {Φn}n=0 of Lμ2 is also a basis of L2. Recall that the set {Φn}n=0 is a basis of L2 if every element ψL2 can be written uniquely in terms of the functions {Φn}n=0. However, in general, this is not true. If s > 0, the Lμ2 basis {Φn}n=0 is overcomplete in L2 (Walter, 1973; Russakovskii, 1997).

2. Uniform convergence and term-by-term differentiability

Along with the eigenfunction expansion (23) in terms of the eigenfunctions {Φn}n=0, we also have the expansion

n=0Ψ,ΦnΦn,Φnϕn
(24)

in terms of the solutions ϕn. The two expansions differ in their behavior at λ-dependent boundaries, z = zi for iS, but are otherwise equal. In particular, the Φn eigenfunction expansion (23) must converge to Ψ(zi) at z = zi for iS as this equality is required for Ψ to be equal to the series expansion (23) in Lμ2 [see the discussion following Eq. (9)]. Some properties of both expansions are given in Appendix A 4. In particular, Theorem A.4 shows that the ϕn solution series (24) does not generally converge to Ψ(zi) at z = zi.

The following theorem is of central concern for physical applications:

Theorem II.5
(uniform convergence). Letψbe a twice continuously differentiable function on [z1, z2] satisfying allλ-independent boundary conditions of the eigenvalue problem(1)–(3). Define the function Ψ onz ∈ [z1, z2] by
Ψ(z)=ciψ(z)di(pψ)(z)atz=zi,foriS,ψ(z)otherwise.
(25)
Then,
ψ(z)=n=0Ψ,ΦnΦn,Φnϕn(z)andψ(z)=n=0Ψ,ΦnΦn,Φnϕn(z),
(26)
with both series converging uniformly and absolutely on [z1, z2].

Proof.

See Appendix B 4.□

If ci = 1 and di = 0 for iS, then we can replace Φn by ϕn and Ψ by ψ in Eq. (26).

In addition, if both boundary conditions of the eigenvalue problem (1)–(3) are λ-dependent, then both expansions in Eq. (26) converge uniformly on [z1, z2] regardless of the boundary conditions ψ satisfies. As discussed in Appendix A 4, for traditional Sturm–Liouville expansions, an analogous result holds only if ψ satisfies the same boundary conditions as the eigenfunctions. Figure 1 contrasts the convergence behavior of such a problem (with continuous eigenfunctions, so ci = 1 and di = 0 for iS) with the convergence behavior of sine and cosine series. All numerical solutions in this article are obtained using a pseudo-spectral code in Dedalus (Burns et al., 2020).

FIG. 1.

Convergence to a function F(z) = 1 + 2z + (3/2)sin(2πz)cos(π2z2 + 3) for z ∈ [−1, 0], shown in panel (a), by various eigenfunction expansions of −ϕ″ = λϕ with fifteen terms, as discussed in Sec. II B 2. Panel (b) shows the Fourier sine expansion of F. Since the sine eigenfunctions vanish at the boundaries z = −1, 0, the series expansion will not converge to F at the boundaries. Panel (c) shows the cosine expansion of F, which converges uniformly to F on the closed interval [−1, 0]. Panel (d) shows an expansion with boundary coefficients in Eqs. (2) and (3) given by (a1, b1, c1, d1) = (−0.5, −5, 1, 0) and (a2, b2, c2, d2) = (0.5, −5, 1, 0). Since ci = 1 and di = 0, then Φn = ϕn and the series expansions (23) and (24) coincide. As with the cosine series, the expansion converges uniformly to F on [−1, 0]. The derivative of F is shown in panel (e). Panel (f) shows the derivative of the sine series expansion. In panel (g), we show the differentiated cosine series that does not converge to the derivative F′ at the boundaries z = z1, z2. In contrast, in panel (h), the differentiated series obtained from a problem with λ-dependent boundary conditions converges uniformly to the derivative F′.

FIG. 1.

Convergence to a function F(z) = 1 + 2z + (3/2)sin(2πz)cos(π2z2 + 3) for z ∈ [−1, 0], shown in panel (a), by various eigenfunction expansions of −ϕ″ = λϕ with fifteen terms, as discussed in Sec. II B 2. Panel (b) shows the Fourier sine expansion of F. Since the sine eigenfunctions vanish at the boundaries z = −1, 0, the series expansion will not converge to F at the boundaries. Panel (c) shows the cosine expansion of F, which converges uniformly to F on the closed interval [−1, 0]. Panel (d) shows an expansion with boundary coefficients in Eqs. (2) and (3) given by (a1, b1, c1, d1) = (−0.5, −5, 1, 0) and (a2, b2, c2, d2) = (0.5, −5, 1, 0). Since ci = 1 and di = 0, then Φn = ϕn and the series expansions (23) and (24) coincide. As with the cosine series, the expansion converges uniformly to F on [−1, 0]. The derivative of F is shown in panel (e). Panel (f) shows the derivative of the sine series expansion. In panel (g), we show the differentiated cosine series that does not converge to the derivative F′ at the boundaries z = z1, z2. In contrast, in panel (h), the differentiated series obtained from a problem with λ-dependent boundary conditions converges uniformly to the derivative F′.

Close modal

Another novel property of the eigenfunction expansions is that we obtain pointwise convergence to functions that are smooth in the interior of the interval, (z1, z2), but have finite jump discontinuities at λ-dependent boundaries (see Appendix A 4). If di ≠ 0 for iS, the convergence is even uniform (Fulton, 1977, Corollary 2.1). Figure 2 illustrates the convergence behavior for eigenfunction expansions with λ-dependent boundary conditions in the two cases di = 0 and di ≠ 0. Note the presence of Gibbs-like oscillations in the case di = 0 shown in panel (b). Although the Φn eigenfunction series (23) converges pointwise to the discontinuous function, the ϕn solution series (24) converges to the values given in Theorem A.4 at the λ-dependent boundaries. The ability of these series expansions to converge to functions with boundary jump discontinuities is related to their ability to expand distributions in the Bretherton (1966)δ-function formulation” of a problem.

FIG. 2.

Convergence to a function F̃ with finite jump discontinuities at the boundaries by two eigenfunction expansions (with λ-dependent boundary conditions) of −ϕ″ = λϕ with fifteen terms, as discussed in Sec. II B 2. The function F̃(z) is defined by F̃(z)=F(z) for z ∈ (z1, z2), where F(z) is the function defined in Fig. 1, F(−1) = 0.5 at the lower boundary, and F(0) = −0.7 at the upper boundary. The function F̃ is shown in panel (a). In panel (b), the boundary coefficients in Eqs. (2) and (3) are given by (a1, b1, c1, d1) = (−0.5, −5, 1, 0) and (a2, b2, c2, d2) = (0.5, −5, 1, 0) as in Fig. 1. In panel (c), the boundary coefficients are (a1, b1, c1, d1) = (−0.5, −5, 1, 0.1) and (a2, b2, c2, d2) = (0.5, −5, 1, −0.1). The Φn expansion (23) and the ϕn expansion (24) are not generally equal at the boundaries z = −1, 0; this figure shows the Φn expansion. The Φn series (23) converges pointwise to F̃ on [−1, 0]; however, the convergence will not be uniform if di = 0 for iS, as in panel (b). The boundary values of the Φn series (23) are shown with a black dot. In panel (b), the eigenfunctions Φn are continuous, and a large number of terms are required for the series to converge to the discontinuous function F̃. Panel (c) shows that the discontinuous eigenfunction Φn have almost converged to F̃—including the jump discontinuities; the black dot in panel (c) overlaps with the gray dots, which represent the boundary values of F̃. Although the ϕn series (24) converges to F̃ in the interior (−1, 0), the ϕn series does not generally converge to F̃ at the boundaries but instead converges to the values given in Theorem A.4.

FIG. 2.

Convergence to a function F̃ with finite jump discontinuities at the boundaries by two eigenfunction expansions (with λ-dependent boundary conditions) of −ϕ″ = λϕ with fifteen terms, as discussed in Sec. II B 2. The function F̃(z) is defined by F̃(z)=F(z) for z ∈ (z1, z2), where F(z) is the function defined in Fig. 1, F(−1) = 0.5 at the lower boundary, and F(0) = −0.7 at the upper boundary. The function F̃ is shown in panel (a). In panel (b), the boundary coefficients in Eqs. (2) and (3) are given by (a1, b1, c1, d1) = (−0.5, −5, 1, 0) and (a2, b2, c2, d2) = (0.5, −5, 1, 0) as in Fig. 1. In panel (c), the boundary coefficients are (a1, b1, c1, d1) = (−0.5, −5, 1, 0.1) and (a2, b2, c2, d2) = (0.5, −5, 1, −0.1). The Φn expansion (23) and the ϕn expansion (24) are not generally equal at the boundaries z = −1, 0; this figure shows the Φn expansion. The Φn series (23) converges pointwise to F̃ on [−1, 0]; however, the convergence will not be uniform if di = 0 for iS, as in panel (b). The boundary values of the Φn series (23) are shown with a black dot. In panel (b), the eigenfunctions Φn are continuous, and a large number of terms are required for the series to converge to the discontinuous function F̃. Panel (c) shows that the discontinuous eigenfunction Φn have almost converged to F̃—including the jump discontinuities; the black dot in panel (c) overlaps with the gray dots, which represent the boundary values of F̃. Although the ϕn series (24) converges to F̃ in the interior (−1, 0), the ϕn series does not generally converge to F̃ at the boundaries but instead converges to the values given in Theorem A.4.

Close modal

Recall that for regular Sturm–Liouville problems [i.e., Eqs. (1)–(3) with ci = di = 0], we obtain a countable infinity of real simple eigenvalues, λn, that may be ordered as

λ0<λ1<λ2<,
(27)

with associated eigenfunctions ϕn. The nth eigenfunction ϕn has n internal zeros in the interval (z1, z2) so that no two eigenfunctions have the same number of internal zeros.

However, once the eigenvalue λ appears in the boundary conditions, there may be up to s + 1 linearly independent eigenfunctions with the same number of internal zeros. The crucial parameters deciding the number of zeros is −bi/di for iS, where bi and di are the boundary coefficients appearing in the boundary conditions (2) and (3). The following lemma outlines the possibilities when only one boundary condition is λ-dependent:

Lemma II.6

(location of double oscillation count). Suppose thats = 1,iS, and letκbe the number of negativeDifor the eigenvalue problem(1)–(3). We have the following possibilities:

  • Right-definite,di ≠ 0: The eigenfunction Φncorresponding to the eigenvalueλnhasninternal zeros ifλn < −bi/diandn − 1 internal zero ifbi/diλn.

  • Right-definite,di = 0: Thenth eigenfunction hasninternal zeros.

  • Left-definite: Ifκ = 0, then all eigenvalues are positive, the problem is right-definite, and either (i) or (ii) applies. Otherwise, ifκ = 1, then the eigenvalues may be ordered as
    λ0<0<λ1<λ2<.
    (28)
    Both eigenfunctions Φ0and Φ1have no internal zeros. The remaining eigenfunctions Φn, forn > 1, haven − 1 internal zeros.

Proof.

Parts (i), (ii), and (ii) are due to Linden (1991); Binding et al. (1994); and Binding and Browne (1999), respectively.□

When both boundary conditions are λ-dependent, the situation is similar. See Binding et al. (1994); Binding and Browne (1999) for further discussion.

Consider a rotating Boussinesq fluid on an f-plane with a reference Boussinesq density of ρ0. The fluid is subject to a constant gravitational acceleration g in the downward, ẑ, direction and to a surface tension T [with dimensions of force per unit length, see Lamb (1975)] at its upper boundary. The upper boundary of the fluid, given by z = η, is a free-surface defined by the function η(x, t), where x=x̂x+ŷy is the horizontal position vector. The lower boundary of the fluid is a flat rigid surface given by z = −H. The fluid region is periodic in both horizontal directions x̂ and ŷ.

The governing equations for infinitesimal perturbations about a background state of no motion, characterized by a prescribed background density of ρB = ρB(z), are

t22w+f02z2w+N2z2w=0forzH,0,
(29)
w=0forz=H,
(30)
t2zwf02zw+gbz2wτz4w=0forz=0,
(31)

where w is the vertical velocity, f0 is the constant value of the Coriolis frequency, the prescribed buoyancy frequency N2 is given by

N2(z)=gρ0dρB(z)dz,
(32)

the acceleration gb is the effective gravitational acceleration at the upper boundary,

gb=gρ0ρaρB(0)
(33)

where ρa is the density of the overlying fluid, and the parameter τ is given by

τ=Tρ0
(34)

where T is the surface tension. The three-dimensional Laplacian is denoted by 2=x2+y2+z2, the horizontal Laplacian is denoted by z2=x2+y2, and the horizontal biharmonic operator is given by z4=z2z2. See Eq. (1.37) in Dingemans (1997) for the surface tension term in (31). The remaining terms in Eqs. (29)–(31) are standard (Gill, 1982). Consistent with our assumption that η(x, t) is small, we evaluate the upper boundary condition at z = 0 in Eq. (31).

We assume wave solutions of the form

w(x,z,t)=ŵ(z)eikxωt,
(35)

where k=x̂kx+ŷky is the horizontal wavevector and ω is the angular frequency. Substituting the wave solution (35) into Eqs. (29)–(31) and setting f0 = 0 yield

ŵ+k2ŵ=σ2N2ŵforz(H,0),
(36)
ŵ=0forz=H,
(37)
(gb+τk2)1ŵ=σ2ŵforz=0,
(38)

where σ = ω/k is the phase speed and k=k is the horizontal wavenumber. Equations (36)–(38) are an eigenvalue problem for the eigenvalue λ = σ−2.

1. Definiteness and the underlying function space

Equations (36)–(38) form an eigenvalue problem with one λ-dependent boundary condition, namely, the upper boundary condition (38). The underlying function space is then

Lμ2L2C.
(39)

We write Ŵn for the eigenfunctions and ŵn for the solutions of the eigenvalue problem (36)–(38) [see the paragraph containing Eq. (22)]. The eigenfunctions Ŵn are related to the solutions ŵn by Eq. (9) with boundary values Ŵn(0) given by Eq. (22). However, since c2 = 1 and d2 = 0 in Eq. (38) [compare with Eqs. (1)–(3)], then Ŵn=ŵn on the closed interval [−H, 0]; thus, the solutions wn are also the eigenfunctions.

By Theorem II.3, the eigenfunctions {ŵn}n=0 form an orthonormal basis of Lμ2. For functions φ and ϕ, the inner product is

φ,ϕ=1N02HH0φϕN2dz+(gb+τk2)φ(0)ϕ(0)
(40)

obtained from Eqs. (4) and (11); we have introduced the factor 1/(N02H) in the above expression for dimensional consistency in eigenfunction expansions (N02 is a typical value of N2). Orthonormality is then given by

δmn=ŵm,ŵn,
(41)

and we have chosen the solutions ŵn to be non-dimensional (so the Kronecker delta is non-dimensional as well).

One verifies that the eigenvalue problem (36)–(38) is right-definite using Proposition II.1 and left-definite using Proposition II.2. Right-definiteness implies that Lμ2, with the inner product (40), is a Hilbert space. That is, all eigenfunctions ŵn satisfy

ŵn,ŵn>0.
(42)

Left-definiteness, along with Proposition II.4, ensures that all eigenvalues λn=σn2 are positive. Indeed, the phase speeds σn satisfy

σ02>σ12>>σn2>0.
(43)

2. Properties of the eigenfunctions

By Lemma II.6, the nth eigenfunction ŵn has n internal zeros in the interval (−H, 0). See Fig. 3 for an illustration of the first six eigenfunctions.

FIG. 3.

The vertical velocity eigenfunctions Ŵn=ŵn of the non-rotating Boussinesq eigenvalue problem (36)–(38) for two distinct wavenumbers with constant stratification, as discussed in Sec. III B. For both wavenumbers, the nth eigenfunction has n internal zeros as in regular Sturm–Liouville theory. The zeroth mode (n = 0) corresponds to a surface gravity wave and is trapped to the upper boundary for large horizontal wavenumbers. In contrast to the internal wave problem with a rigid lid, the modes ŵn now depend on the horizontal wavenumber k through the boundary condition (38); however, this dependence is weak for n ≫ 1, as can be observed in this figure; for n > 2, the modes for k = 0.01 (in black) and for k = 10 (in gray) nearly coincide. The horizontal wavenumbers k are non-dimensionalized by H.

FIG. 3.

The vertical velocity eigenfunctions Ŵn=ŵn of the non-rotating Boussinesq eigenvalue problem (36)–(38) for two distinct wavenumbers with constant stratification, as discussed in Sec. III B. For both wavenumbers, the nth eigenfunction has n internal zeros as in regular Sturm–Liouville theory. The zeroth mode (n = 0) corresponds to a surface gravity wave and is trapped to the upper boundary for large horizontal wavenumbers. In contrast to the internal wave problem with a rigid lid, the modes ŵn now depend on the horizontal wavenumber k through the boundary condition (38); however, this dependence is weak for n ≫ 1, as can be observed in this figure; for n > 2, the modes for k = 0.01 (in black) and for k = 10 (in gray) nearly coincide. The horizontal wavenumbers k are non-dimensionalized by H.

Close modal

The eigenfunctions {ŵn}n=0 are complete in L2 but do not form a basis in L2; in fact, the basis is overcomplete in L2. The presence of a free-surface provides an additional degree of freedom over the usual rigid-lid L2 basis of internal wave eigenfunctions. Indeed, the n = 0 wave in Fig. 3 corresponds to a surface gravity wave, while the remaining modes are internal gravity waves (with some surface motion).

3. Expansion properties

Given a twice continuously differentiable function χ(z) satisfying χ(−H) = 0, then from Theorem II.5, we have

χ(z)=n=0χ,ŵnŵn(z)andχ(z)=n=0χ,ŵnwn(z),
(44)

with both series converging uniformly on [−H, 0] (note that χ is not required to satisfy any particular boundary condition at z = 0). If χ is the vertical structure at time t = 0 (and at some wavevector k) and we assume ∂tw(x, z, t = 0) = 0, then the subsequent time-evolution is given by

w(x,z,t)=n=0χ,ŵnwn(z)cosσnkteikx.
(45)

4. The f-plane hydrostatic problem

Suppose we have hydrostatic gravity waves on an f-plane with free surface at the upper boundary, as in Kelly (2016). The appropriate inner product is obtained by setting τ = 0 in the inner product (40). All the above results on the eigenfunctions of gravity-capillary waves carry over to the hydrostatic f-plane problem, provided we set

σ2=ω2f02k2.
(46)

Although this next example is not geophysically relevant, it has the curious property that the resulting eigenfunctions are discontinuous.

Let N02 be a typical value of N2(z). Consider the situation where f02/N021 but

gb+τk2f02HO(1).
(47)

Accordingly, we may neglect the Coriolis parameter in the interior equation (29) but not at the upper boundary condition (31). Substituting the wave solution (35) into Eqs. (29)–(31) yields

ŵ+k2ŵ=σ2N2ŵforz(H,0),
(48)
ŵ=0forz=H,
(49)
(gb+τk2)1ŵ=σ2ŵ+f02k2(gb+τk2)1ŵforz=0,
(50)

where σ = ω/k is the phase speed. Equations (48)–(50)form an eigenvalue problem for the eigenvalue λ = σ−2.

1. Definiteness and the underlying function space

As in the previous case, the eigenvalue problem is both right-definite and left-definite, the underlying function space Lμ2 is given by Eq. (39), and the appropriate inner product is Eq. (40). By right-definiteness, the space Lμ2, equipped with the inner product (40), is a Hilbert space; thus, all eigenfunctions Ŵn satisfy

Ŵm,Ŵn>0.
(51)

By Theorem II.3, all eigenvalues λn=σn2 are real and the corresponding eigenfunctions {Ŵn}n=0 form an orthonormal basis of the Hilbert space Lμ2. By Proposition II.4, all eigenvalues λn=σn2 are positive and satisfy Eq. (43).

2. Boundary jump discontinuity of the eigenfunctions

The main difference between the previous non-rotating problem (36)–(38) and the above problem (48)–(50) is that, in the present problem, if f0 ≠ 0, then d2 ≠ 0 [see Eq. (3)]. Thus, by Eq. (22), the eigenfunctions Ŵn generally have a jump discontinuity at the upper boundary z = 0 (see Fig. 4) and so are not equal to the solutions ŵn. The eigenfunctions Ŵn are defined by Ŵn(z)=ŵn(z) for z ∈ [−H, 0) and

Ŵn(0)=ŵn(0)+f02k2(gb+τk2)1ŵn(0)
(52)

[see Eq. (22)]. It is not difficult to show that

Ŵn(0)0forn sufficiently large,
(53)

as can be seen in Fig. 4.

FIG. 4.

The vertical velocity eigenfunctions Ŵn of a Boussinesq fluid with a rotating upper boundary—eigenvalue problem (48)–(50). This figure is discussed in Sec. III C. The wavenumbers k in the figure are non-dimensionalized by the depth H. The dots represent the values of the eigenfunctions at the boundaries. Note that the eigenfunctions have a finite jump discontinuity at z = 0. For kH = 0.01 (given by the black line), there are two modes with no internal zeros. As k increases, we obtain two modes with one internal zero (at kH = 0.05, the thick gray line) and then two modes with three internal zeros (at kH = 0.11, the thin gray line).

FIG. 4.

The vertical velocity eigenfunctions Ŵn of a Boussinesq fluid with a rotating upper boundary—eigenvalue problem (48)–(50). This figure is discussed in Sec. III C. The wavenumbers k in the figure are non-dimensionalized by the depth H. The dots represent the values of the eigenfunctions at the boundaries. Note that the eigenfunctions have a finite jump discontinuity at z = 0. For kH = 0.01 (given by the black line), there are two modes with no internal zeros. As k increases, we obtain two modes with one internal zero (at kH = 0.05, the thick gray line) and then two modes with three internal zeros (at kH = 0.11, the thin gray line).

Close modal

Physical motion is given by the solutions ŵn, which are continuous over the closed interval [−H, 0]. The jump discontinuity in the eigenfunctions Ŵn does not correspond to any physical motion; instead, the eigenfunctions Ŵn are convenient mathematical aids used to obtain eigenfunction expansions in the function space Lμ2.

3. Number of internal zeros of the eigenfunctions

Another consequence of d2 ≠ 0 is that by, Lemma II.6, there are two distinct solutions ŵM and ŵM+1 with the same number of internal zeros (i.e., M) in the interval (−H, 0). Noting that

b2d2=k2f02,
(54)

the integer M is determined by

σ02>σ12>>σM2>f02k2σM+12>>0.
(55)

A smaller f0 or a larger k implies a larger M and hence that ŵM and ŵM+1 have a larger number of internal zeros, as shown in Fig. 4.

4. Expansion properties

As in the previous problem, the eigenfunctions are complete in Lμ2 but overcomplete in L2 due to the additional surface gravity-capillary wave.

Given a twice continuously differentiable function χ(z) satisfying χ(−H) = 0, we define the discontinuous function X(z) by

X(z)=χ(z)forz[H,0),χ(0)+f02k2gb+τk21χ(0)forz=0,
(56)

as in Theorem II.5. Then, by Theorem II.5, we have the expansions

χ(z)=n=0X,Ŵnŵn(z)andχ(z)=n=0X,Ŵnwn(z).
(57)

Moreover, if χ(z) is the vertical structure at t = 0 (and at some wavevector k) and we assume ∂tw(x, z, t = 0) = 0, then the subsequent time-evolution is given by

w(x,z,t)=n=0X,Ŵnŵn(z)cosσnkteikx.
(58)

Linearizing the quasigeostrophic equations about a quiescent background state with an infinitesimally sloping lower boundary, at z = −H, and a rigid flat upper boundary, at z = 0, renders

tz2ψ+zS1zψ+ẑzψ×zf=0forz(H,0),
(59)
tS1zψ+ẑzψ×f0zh=0forz=H,
(60)
tS1zψ=0forz=0.
(61)

See Rhines (1970); Charney and Flierl (1981); and Straub (1994) for details. The streamfunction ψ is defined through u=ẑ×zψ, where u is the horizontal velocity and z=x̂x+ŷy is the horizontal Laplacian. The stratification parameter S is given by

S(z)=N2(z)f02,
(62)

where N2 is the buoyancy frequency and f0 is the reference Coriolis parameter. The latitude dependent Coriolis parameter f is defined by

f(y)=f0+βy.
(63)

Finally, h(x) is the height of the topography at the lower boundary and is a linear function of the horizontal position vector x. Consistent with quasigeostrophic theory, we assume that topography h is small, and so we evaluate the lower boundary condition at z = −H in Eq. (60).

We assume wave solutions of the form

ψ(x,z,t)=ψ̂(z)ei(kxωt),
(64)

where k=x̂kx+ŷky is the horizontal wavevector and ω is the angular frequency.

We denote by Δθf the angle between the horizontal wavevector k and the gradient of Coriolis parameter zf,

sin(Δθf)=1kβẑk×zf,
(65)

where k=k is the horizontal wavenumber. Positive angles are measured counter-clockwise relative to k. Thus, Δθf > 0 indicates that k points to the right of zf, while Δθf < 0 indicates that k points to the left of zf.

We define the topographic parameter α by

α=f0zh.
(66)

In analogy with Δθf, we define the angle Δθh by

sin(Δθh)=1kαẑk×f0zh
(67)

with a similar interpretation assigned to Δθh > 0 and Δθh < 0.

Substituting the wave solution (64) into the linear quasigeostrophic equations (59)–(61) and assuming that α sin(Δθh) ≠ 0, ω ≠ 0, and k ≠ 0, we obtain

(S1ψ̂)+k2ψ̂=λψ̂forz(H,0),
(68)
βαsin(Δθf)sin(Δθh)S1ψ̂=λψforz=H,
(69)
S1ψ=0forz=0,
(70)

where we have defined the eigenvalue λ by

λ=kβsin(Δθf)ω.
(71)

Since k ≠ 0, then λ = 0 is not an eigenvalue. The above problem (68)–(70) was recently considered in LaCasce (2017).

1. Definiteness and the underlying function space

The eigenvalue problem has one λ-dependent boundary condition, and so the underlying function space is

Lμ2L2C.
(72)

The appropriate inner product is obtained from Eqs. (4) and (11),

φ,ϕ=1HH0φϕdz+αβsinΔθhsinΔθfφ(H)ϕ(H),
(73)

where we have introduced the factor 1/H for dimensional consistency in eigenfunction expansions. By Proposition II.1, the problem is right-definite for horizontal wavevectors k satisfying

sin(Δθh)sin(Δθf)>0,
(74)

and in such cases, Lμ2 equipped with the inner product (11) is a Hilbert space. However, Lμ2 is not a Hilbert space for all wavevectors k. By Proposition II.2, the problem is left-definite for all wavevectors k, and so Lμ2, equipped with the inner product (11), is generally a Pontryagin space.

We write Ψ̂n for the eigenfunctions and ψ̂n for the solutions of Eqs. (68)–(70). The eigenfunctions Ψ̂n are related to the solutions ψ̂n by (9) with boundary values Ψ̂n(0) given by Eq. (22). However, since c1 = 1 and d1 = 0 in Eq. (69) [compare with Eqs. (1)–(3)], then Ψ̂n=ψ̂n on the closed interval [−H, 0]. Thus, the solutions ψn are also the eigenfunctions.

With Theorem II.3, we deduce that all eigenvalues λn are real and the corresponding eigenfunctions {ψ̂n}n=0 form an orthonormal basis for Lμ2. Orthonormality is defined with respect to the inner product given by Eq. (73) and takes the form

±δmn=ψ̂m,ψ̂n,
(75)

where we have taken the eigenfunctions ψ̂m and ψ̂n to be non-dimensional.

2. Properties of the eigenfunctions

By Lemma II.6, the number of internal zeros of the eigenfunctions {ψ̂n}n=0 depends on the propagation direction and hence [by Eq. (74)] on the definiteness of the problem (see Fig. 5):

  1. If the problem is right-definite, then the nth eigenfunction has n internal zeros.

  2. If the problem is not right-definite, then both ψ0 and ψ1 have no internal zeros; the remaining eigenfunctions ψn, for n > 1, have n − 1 internal zeros.

FIG. 5.

The streamfunction eigenfunctions ψ̂n of the quasigeostrophic eigenvalue problem with a sloping bottom from Sec. IV B. Two cases are shown. The first is with Δθf = −90° and Δθ1 = −30° and is both right-definite and left-definite. The second is with Δθf = −45° and Δθ1 = 15° and is only left-definite. In the right-definite case, the nth eigenfunction has n internal zero, whereas in the left-definite only case, there are two eigenfunctions (n = 0, 1) with no internal zeros.

FIG. 5.

The streamfunction eigenfunctions ψ̂n of the quasigeostrophic eigenvalue problem with a sloping bottom from Sec. IV B. Two cases are shown. The first is with Δθf = −90° and Δθ1 = −30° and is both right-definite and left-definite. The second is with Δθf = −45° and Δθ1 = 15° and is only left-definite. In the right-definite case, the nth eigenfunction has n internal zero, whereas in the left-definite only case, there are two eigenfunctions (n = 0, 1) with no internal zeros.

Close modal

As the problem is left-definite for all wavevectors k, we can use Proposition II.4 to determine the sign of the eigenvalues. Proposition II.4 informs us that

λnψ̂n,ψ̂n>0.
(76)

In the first case, when the problem is right-definite, all eigenvalues are positive and all eigenfunctions ψ̂n satisfy ψ̂n,ψ̂n>0. In the second case, when the problem is only left-definite, then there is one negative eigenvalue λ0 and the corresponding eigenfunction ψ̂0 satisfies ψ̂0,ψ̂0<0. The remaining eigenvalues are positive, and their corresponding eigenfunctions satisfy ψ̂n,ψ̂n>0. In fact, from Eq. (71), we see that waves with ψ̂n,ψ̂n>0 have westward phase speeds ωn/k < 0, while waves with ψ̂n,ψ̂n<0 have eastward phase speeds ωn/k > 0.

3. Expansion properties

The eigenfunctions {ψ̂n}n=0 are complete in Lμ2 but overcomplete in L2. Physically, there is now an additional eigenfunction corresponding to a topographic Rossby wave (n = 0 in Fig. 5).

Given a twice continuously differentiable function ϕ(z) satisfying ϕ′(0) = 0, then from Theorem II.5, we have

ϕ(z)=n=0ϕ,ψ̂nψ̂n,ψ̂nψ̂n(z)andϕ(z)=n=0ϕ,ψ̂nψ̂n,ψ̂nψ̂n(z),
(77)

with both series converging uniformly on [−H, 0] (note that ϕ is not required to satisfy any particular boundary condition at z = −H). If the vertical structure at time t = 0 (and at some wavevector k) is given by ϕ, then the subsequent time-evolution is given by

ψ(x,z,t)=n=0ϕ,ψ̂nψ̂n,ψ̂nψ̂n(z)cosωnteikx,
(78)

where the angular frequency ωn is given by Eq. (71).

We now consider a localized perturbation at a dynamically active boundary; we idealize such a perturbation by a boundary step-function Θi (for iS) given by

Θi(z)=1ifz=zi,0otherwise.
(79)

Using Eq. (23), the series expansion of Θi is found to be

Θi=1Din=0Φn(zi)Φn,ΦnΦn(z).
(80)

For the non-rotating Boussinesq problem of Sec. III B, a step-function perturbation with amplitude w0 (at some wavevector k) yields the time-evolution

w(x,z,t)=w0gb+τk2N02Hn=0ŵn(0)ŵn(z)cosσnkteikx.
(81)

Analogously, for the quasigeostrophic problem of Sec. IV B, a step-function perturbation with amplitude ψ0 (at some wavevector k) yields the time-evolution

ψ(x,z,t)=ψ0αsin(Δθh)Hβsin(Δθf)n=0ψ̂n(H)ψ̂n,ψ̂nψ̂n(z)cosωnteikx.
(82)

That both the above series converge to a step-function at t = 0 (and x = 0) is confirmed by Theorem A.3 along with Theorem 2 in Fulton (1977).

We thus see that a step-function perturbation induces wave motion with an amplitude that is proportional to the boundary-confined restoring force (at wavevector k). Moreover, the amplitude of each constituent wave in the resulting motion is proportional to the projection of that wave onto the dynamically active boundary.

We have developed a mathematical framework for the analysis of three-dimensional wave problems with dynamically active boundaries (i.e., boundaries where time derivatives appear in the boundary conditions). The resulting waves have vertical structures that depend on the wavevector k: For Boussinesq gravity waves, the dependence is only through the wavenumber k, whereas the dependence for quasigeostrophic Rossby waves is on both the wavenumber k and the propagation direction k/k. Moreover, the vertical structures of the waves are complete in a space larger than L2, namely, they are complete in Lμ2L2Cs, where s is the number of dynamically active boundaries (and the number of boundary-trapped waves). Each dynamically active boundary contributes an additional boundary-trapped wave and hence an additional degree of freedom to the problem. Mathematically, the presence of boundary-trapped waves allows us to expand a larger collection of functions (with a uniformly convergent series) in terms of the modes. The resulting series are term-by-term differentiable and the differentiated series converges uniformly. In fact, the normal modes have the intriguing property converging pointwise to functions with finite jump discontinuities at the boundaries, a property related to their ability to expand distributions in the Bretherton (1966)δ-function formulation” of a physical problem. By considering a step-function perturbation at a dynamically active boundary, we find that the subsequent time-evolution consists of waves whose amplitude is proportional to their projection at the dynamically active boundary. Within the mathematical formulation is a qualitative oscillation theory relating the number of internal zeros of the eigenfunctions to physical quantities; indeed, for the quasigeostrophic problem, the number of zeros of the topographic Rossby wave depends on the propagation direction, while, for the rotating Boussinesq problem, the ratio of the Coriolis parameter to the horizontal wavenumber determines at which integer M we obtain two modes with M zeros.

Our results also clarify the difference between the traditional quasigeostrophic baroclinic modes and the L2C2 eigenfunctions of Smith and Vanneste (2012). In particular, the series expansion of a function in terms of the Smith and Vanneste (2012) eigenfunctions has a term-by-term derivative that converges uniformly over the whole interval regardless of the boundary conditions satisfied by the function. In contrast, an eigenfunction expansion in terms of the baroclinic modes only has this property if the function satisfies the same boundary conditions as the baroclinic modes. One consequence is the following. Suppose we expand an arbitrary quasigeostrophic state, with boundary buoyancy anomalies, in terms of the baroclinic modes. The presence of these boundary buoyancy anomalies implies that this state does not satisfy the same boundary conditions as the baroclinic modes. The resulting series expansion in term of the baroclinic modes is then not differentiable at the boundaries. We are thus unable to recover the value of the boundary buoyancy anomalies from the series expansion, and so we have lost information in the expansion process. This loss of information does not occur with L2C2 expansions.

Normal mode decompositions of quasigeostrophic motion play an important role in physical oceanography [e.g., Wunsch (1997); Lapeyre (2009); and LaCasce (2017)]. Other applications include the extension of equilibrium statistical mechanical calculations [e.g., Bouchet and Venaille (2012); Venaille et al. (2012)] to three-dimensional systems with dynamically active boundaries. Moreover, the mathematical framework developed here is useful for the development of weakly non-linear wave turbulence theories [e.g., Fu and Flierl (1980); Smith and Vallis (2001); and Scott (2014)] in systems with both internal and boundary-trapped waves.

H.Y. is grateful to Stephen Griffies for numerous readings, helpful comments, and advice offered throughout this project. This report was prepared by H.Y. under Award No. NA18OAR4320123 from the National Oceanic and Atmospheric Administration, U.S. Department of Commerce. The statements, findings, conclusions, and recommendations are those of the author and do not necessarily reflect the views of the National Oceanic and Atmospheric Administration, or the U.S. Department of Commerce.

The data that support the findings of this study are available within the article and its supplementary material.

1. Construction of Lμ2

First, define the weighted Lebesgue measure σ by

σ([a,b])=abrdzwhere a,b[z1,z2].
(A1)

The measure σ induces the differential element

dσ(z)=r(z)dz
(A2)

and is the measure associated with L2 [see Eqs. (6) and (7)].

Now, for iS [see Eq. (8)], define the pure point measure νi by [e.g., Reed and Simon (1980), Sec. I D, Example 2]

νi([a,b])=Di1ifzi[a,b],0otherwise,
(A3)

where Di is the combination of boundary condition coefficients given by Eq. (4). The pure point measure νi induces the differential element

dνi(z)=Di1δ(zzi)dz,
(A4)

where δ(z) is the Dirac distribution.

Consider now the space Lνi2 of “functions” ϕ satisfying

z1z2ϕ2dνi=Di1z1z2ϕ2δ(zzi)dz=Di1ϕ(zi)2<.
(A5)

Elements of Lνi2 are not functions but rather equivalence classes of functions. Two functions, ϕ and ψ, on the interval [z1, z2] are equivalent in Lνi2 if ϕ(zi) = ψ(zi). In particular, Lνi2 is a one-dimensional vector space and is hence isomorphic to the field of complex numbers C,

Lνi2C.
(A6)

Now, define the measure μ by

μ=σ+iSνi
(A7)

with an induced differential element of

dμ(z)=r(z)+iSDi1δ(zzi)dz.
(A8)

Then, Lμ2 is the space of equivalence classes of functions that are square-integrable with respect to the measure μ.

Since the measures σ and νi, for iS, are mutually singular, we have [Reed and Simon (1980), Sec. II A, Example 5]

Lμ2L2iSLνi2L2Cs
(A9)

from which we see that Lμ2 is “larger” by s dimensions.

2. The eigenvalue problem in Lμ2

We construct here an operator formulation of (1)–(3) as an eigenvalue problem in the Pontryagin space Lμ2.

Define the differential operator acting on a function ϕ by

ϕ=1r(pϕ)qϕ.
(A10)

We also define the following boundary operators for iS:

Biϕ=aiϕ(zi)bi(pϕ)(zi),
(A11)
Ciϕ=ciϕ(zi)di(pϕ)(zi).
(A12)

Let Φ be an element of Lμ2, as in Eq. (9), with boundary values Φ(zi)=Ciϕ for iS and equal to ϕ elsewhere. We then define the operator L, acting on functions Φ, by

LΦ=ϕforz(z1,z2),Biϕforz=zi where iS
(A13)

with a domain D(L)Lμ2 defined by

D(L)={ΦLμ2|ϕ is continuously differentiable, ϕL2,Φ(zi)=CiϕforiS and Biϕ=0fori{1,2}\S}.
(A14)

Recall that S contains indices of the λ-dependent boundary conditions, and therefore, {1, 2}\S contains the indices of the λ-independent boundary conditions.

Then, on the subspace D(L) of Lμ2, the eigenvalue problem (1)–(3) may be written as

LΦ=λΦ.
(A15)

As shown in Russakovskii (1975; 1997), L is a self-adjoint operator in the space Lμ2.

There is a natural quadratic form Q, induced by the eigenvalue problem (1)–(3), given by

Q(Φ,Ψ)=Φ,LΨ.
(A16)

For elements Φ,ΨD(L), we obtain

Q(Φ,Ψ)=z1z2pϕ*ψ+qϕ*ψdz+i{1,2}\S(1)i+1aibiϕ(zi)*ψ(zi)iS1Diψ(zi)(pψ)(zi)*aiciaidiaidibidiϕ(zi)(pϕ)(zi)
(A17)

for bi ≠ 0 for i ∈ {1, 2}\S. If bi = 0 for i ∈{1, 2}\S, then we replace the term ai/bi with zero.

To develop the reality and completeness theorem (Theorem II.3), we provide the following definitions:

Definition A.1
(right-definite). The eigenvalue problem(1)–(3)is said to be right-definite ifLμ2is a Hilbert space or, equivalently, if
Φ,Φ>0
(A18)
for all non-zeroΦLμ2.

Definition A.2
(left-definite). The eigenvalue problem(1)–(3)is said to be left-definite if
Q(Φ,Φ)0
(A19)
for allΦD(L).

One can then prove Propositions II.1 and II.2 through straightforward manipulations.

3. Properties of eigenfunction expansions

The following theorem features some of the novel properties of the basis {Φn}n=0 of Lμ2. Theorem A.3 is a generalization of a theorem first formulated, in the right-definite case, by Walter (1973); Fulton (1977).

Theorem A.3

(eigenfunction expansions). Let{Φn}n=0be the set of eigenfunctions of the eigenvalue problem(1)–(3). Then, the following properties hold.

  • Null series: ForiS, we have
    0=Di1n=01Φn,ΦnΦn(zi)ϕn(z)
    (A20)
    with equality in the sense ofL2.
  • Unit series: ForiS, we have
    1=Di1n=01Φn,ΦnΦn(zi)2.
    (A21)
  • L2-expansion: LetψL2, then
    ψ=n=01Φn,Φnz1z2ψ*ϕnrdzϕn
    (A22)
    with equality in the sense ofL2.
  • Interior-boundary orthogonality: LetψL2, then foriS, we have
    0=n=01Φn,Φnz1z2ψ*ϕnrdzΦn(zi).
    (A23)

Proof.

The proof is similar to the proof of Corollary 1.1 in Fulton (1977).□

4. Pointwise convergence and Sturm–Liouville series

Theorem 3 in Fulton (1977) states that the Φn series expansion (23) behaves like a Fourier series in the interior of the interval (z1, z2) (see  Appendix B for why this theorem applies in the left-definite case). Since the expansions (23) and (24) in terms of Φn and ϕn are equal in the interior, then the above theorem applies to the ϕn series (24) as well. It is at the boundaries points, z = z1, z2, where the novel behavior of the series expansions (23) and (24) appears.

For traditional Sturm–Liouville expansions [with eigenfunctions of problem (1)–(3) with ci, di = 0 for i = 1, 2], eigenfunction expansions behave like the analogous Fourier series on [z1, z2] [p. 16 in Titchmarsh (1962) or Chap. 1, Sec. 9, in Levitan and Sargsjan (1975)]. In particular, for a twice continuously differentiable function ψ, the eigenfunction expansion of ψ converges uniformly to ψ on [z1, z2] so long as the eigenfunctions ϕn do not vanish at the boundaries. If the eigenfunctions vanish at one of the boundaries, then we only obtain uniform convergence if ψ vanishes at the corresponding boundary as well (Brown and Churchill, 1993, Sec. 22). Under these conditions, the resulting expansion will be differentiable in the interior of the interval, (z1, z2), but not at the boundaries z = z1, z2 [see Chap. 8, Sec. 3, in Levitan and Sargsjan (1975) for the equiconvergence of differentiated Sturm–Liouville series with Fourier series and see Sec. 23 in Brown and Churchill (1993) for the convergence behavior of differentiated Fourier series].

Returning to the case of eigenfunction expansions for the eigenvalue problem (1)–(3) with λ-dependent boundaries, the following theorem provides pointwise (as well as uniform, in the case di ≠ 0) convergence conditions for the ϕn series (24):

Theorem A.4
(pointwise convergence). Letψbe a twice continuously differentiable function on the interval [z1, z2] satisfying anyλ-independent boundary conditions in the eigenvalue problem(1)–(3). Define the function Ψ on [z1, z2] by
Ψ(z)=Ψ(zi)atz=zi,foriS,ψ(z)otherwise,
(A24)
where Ψ(zi) are constants foriS(theλ-dependent boundaries). Then, we have the following:
  • Ifdi ≠ 0 foriS, then theϕnseries expansion(24)converges uniformly toψ(z) on the closed interval [z1, z2],
    n=0Ψ,ΦnΦn,Φnϕn(z)=ψ(z).
    (A25)
    Furthermore, for the differentiated series, we have
    n=0Ψ,ΦnΦn,Φnϕn(z)=ciψ(zi)Ψ(zi)/diatz=ziforiSψ(z)otherwise.
    (A26)
  • Ifdi = 0, then we have
    n=0Ψ,ΦnΦn,Φnϕn=Ψ(zi)/ciatz=ziforiSψ(z)otherwise.
    (A27)

Proof.

This theorem is a generalization of Corollary 2.1 in Fulton (1977). We provide the extension of the corollary to the left-definite problem in Appendix B 4.□

The Φn series (23) converges to Ψ(zi) at z = zi for iS (i.e., at λ-dependent boundaries) but otherwise behaves as in Theorem A.4.

1. Literature survey

There is an extensive literature associated with the eigenvalue problem (1)–(3) with λ-dependent boundary conditions (see Schäfke and Schneider (1966); Fulton (1977), and citations within). One can use the S-Hermitian theory of Schäfke and Schneider (1965; 1966; and 1968) to show that one obtains real eigenvalues when the problem is either right-definite or left-definite (see Sec. II), but completeness results in Lμ2 are unavailable in this theory.

The right-definite theory is well-known (Evans, 1970; Walter, 1973; and Fulton, 1977). In particular, Fulton (1977) applies the residue calculus techniques of Titchmarsh (1962) to the right-definite problem and, in the process, extends some well-known properties of Fourier series to eigenfunction expansions associated with (1)–(3). A recent Hilbert space approach to the right-definite problem, in the context of obtaining a projection basis for quasigeostrophic dynamics, is given by Smith and Vanneste (2012).

The left-definite problem is less examined. As we show in this article, the eigenvalue problem is naturally formulated in a Pontryagin space, and in such a setting, one can prove, in the left-definite case, that the eigenvalues are real and that the eigenfunctions form a basis for the underlying function space. We prove this result, stated in Theorem II.3, in Appendix B 3.

With these completeness results, we may apply the residue calculus techniques of Titchmarsh (1962) to extend the results of Fulton (1977) to the left-definite problem. Indeed, Fulton (1977) used a combination of Hilbert space methods as well as residue calculus techniques to prove various convergence results for the right-definite problem. However, only Theorem 1 of Fulton (1977) makes use of Hilbert space methods. If we extend Fulton’s Theorem 1 to the left-definite problem, then all the results of Fulton (1977) will apply equally to the left-definite problem. A left-definite analog of Theorem 1 of Fulton (1977), along with its proof, is given in Appendix B 4.

2. A Pontryagin space theorem

A Pontryagin space Πκ, for a finite non-negative integer κ, is a Hilbert space with a κ-dimensional subspace of elements satisfying

ϕ,ϕ<0.
(B1)

An introduction to the theory of Pontryagin spaces can be found in Iohvidov and Krein (1960) as well as in the monograph of Bognár (1974). Another resource is the monograph of Azizov and Iokhvidov (1989) on linear operators in indefinite inner product spaces.

Pontryagin spaces admit a decomposition

Πκ=Π+Π
(B2)

into orthogonal subspaces (Π+,+,) and (Π,,). Moreover, one can associate with a Pontryagin space (Πκ,,) a corresponding Hilbert space (Π,,+), where the positive-definite inner product ,+ is defined by

ϕ,ψ+=ϕ+,ψ+ϕ,ψ,ϕ,ψΠ,
(B3)

where ϕ = ϕ+ + ϕ and ψ = ψ+ + ψ, with ϕ±, ψ±∈Π± (Azizov and Iokhvidov, 1981).

As a prerequisite to proving Theorem II.3, we require the following theorem:

Theorem B.1

(positive compact Pontryagin space operators). LetAbe a positive compact operator in a Pontryagin space Πκand suppose thatλ = 0 is not an eigenvalue. Then, all eigenvalues are real and the corresponding eigenvectors form an orthonormal basis for Πκ. There are preciselyκnegative eigenvalues and the remaining eigenvalues are positive. Moreover, positive eigenvalues have positive eigenvectors and negative eigenvalues have negative eigenvectors.

Proof.

By Theorem VII.1.3 in Bognár (1974), the eigenvalues are all real. Moreover, since λ = 0 is not an eigenvalue, then all eigenspaces are definite (Bognár, 1974, Theorem VII.1.2), and hence, all eigenvalues are semi-simple (Bognár, 1974, Lemma II.3.8).

Since A is a compact operator and λ = 0 is not an eigenvalue, then the span of the generalized eigenspaces is dense in Πκ (Azizov and Iokhvidov, 1989, Lemma 4.2.14). Since all eigenvalues are semi-simple, then all generalized eigenvectors are eigenvectors and so the span of the eigenvectors is dense in Πκ. Orthogonality of eigenvectors can be shown as in a Hilbert space.

Let λ be an eigenvalue and ϕ be the corresponding eigenvector. By the positivity of A, we have
Aϕ,ϕ=λϕ,ϕ0.
(B4)
Since all eigenspaces are definite, it follows that positive eigenvectors must correspond to positive eigenvalues and negative eigenvectors must correspond to negative eigenvalues.

Finally, by Theorem IX.1.4 in Bognár (1974), any dense subset of Πκ must contain a negative-definite κ dimensional subspace. Consequently, there are κ negative eigenvectors and hence κ negative eigenvalues.□

3. Proof of theorem II.3

Proof.

The proof for the left-definite case is essentially the standard proof [e.g., Debnath and Mikusinski (2005), Sec. 5.10] with Theorem B.1 substituting for the Hilbert–Schmidt theorem. We give a general outline nonetheless.

First, it is well-known that L is self-adjoint in Lμ2 [e.g., Russakovskii (1975; 1997)]. Since λ = 0 is not an eigenvalue, then the inverse operator L1 exists and is an integral operator on Lμ2. For an explicit construction, see Sec. 4 in Walter (1973); Fulton (1977); and Hinton (1979). The eigenvalue problem for L, Eq. (14), is then equivalent to
L1ϕ=λ1ϕ,
(B5)
and both problems have the same eigenfunctions.

The operator L1 is a positive compact operator and so satisfies the requirements of Theorem B.1. Application of Theorem B.1 to L1 then assures that all eigenvalues λn are real, the eigenfunctions form an orthonormal basis for Lμ2, and the sequence of eigenvalues {λn}n=0 is countable and bounded from below.

The claim that the eigenvalues are simple is verified in Binding and Browne (1999) for the left-definite problem. Alternatively, an argument similar to that of Fulton (1977) and Titchmarsh (1962), (p. 12) can be made to prove the simplicity of the eigenvalues.□

4. Extending Fulton (1977) to the left-definite problem

The following is a left-definite analog of Theorem 1 in Fulton (1977). The proof is almost identical to the right-definite case (Fulton, 1977; Hinton, 1979) with minor modifications. Essentially, since Ψ,Ψ can be negative, we must replace these terms in the inequalities below with the induced Hilbert space inner product Ψ,Ψ+ given by Eq. (B3). Our Lμ2 Green’s functions G corresponds to G̃ in Hinton (1979).

Theorem B.2
(a left-definite extension of Fulton’s Theorem 1). LetΨLμ2be defined on the interval [z1, z2] by
Ψ(z)=Ψ(zi)atz=zi,foriS,ψ(z)otherwise,
(B6)
whereψL2and Ψ(zi) are constants foriS. The eigenfunctions Φnare defined similarly (see2Sec. II).
  • Parseval formula: ForΨLμ2, we have
    Ψ,Ψ=n=0Ψ,Φn2Φn,Φn.
    (B7)
  • ForΨD(L), we have
    Ψ=n=0Ψ,ΦnΦn,ΦnΦn.
    (B8)
    with equality in the sense ofLμ2. Moreover, we have
    ψ=n=0Ψ,ΦnΦn,Φnϕn,
    (B9)
    which converges uniformly and absolutely forz ∈ [z1, z2] and may be differentiated term-by-term, with the differentiated series converging uniformly and absolutely toψforz ∈ [z1, z2]. The boundaries series
    Ψ(zi)=n=0Ψ,ΦnΦn,ΦnΦn(zi)
    (B10)
    foriSis absolutely convergent.

Proof.

The Parseval formula (B7) is a consequence of the completeness of the eigenfunctions {Φn}n=0 in Lμ2, given by Theorems II.3 and IV.3.4 in Bognár (1974). Similarly, expansion (B8) is also due to completeness of the eigenfunctions.

We first prove that the series (B9) converges uniformly and absolutely for z ∈ [z1, z2]. We begin with the identity
ϕn(z)=(λλn)G(z,,λ),Φn,
(B11)
where λC is not an eigenvalue of L and G is the Lμ2 Green’s function [see Eq. (8) in Hinton (1979)]. Then,
n=0λnϕn2λλn2=n=0λnG(z,,λ),Φn2G(z,,λ),LG(z,,λ)+B1(λ),
(B12)
where ,+ is the induced Hilbert space inner product given by Eq. (B3) and B1(λ) is a z independent upper bound [Eq. (9) in Hinton (1979)]. In addition, since ΨD(L), then LΨ,LΨ+<. Thus, we obtain
nλn2Ψ,Φn2=LΨ,LΨ+<.
(B13)
The uniform and absolute convergence of (B9) follows from
n=0Ψ,ΦnΦn,Φnϕn=n=0ϕnλλnλλnΨ,ΦnΦn,Φn
(B14)
n=0ϕnλλn2n=0λλn2Ψ,Φn2
(B15)
along with Eqs. (B12) and (B13). The absolute convergence of the boundary series (B10) follows as well.
To show that the series (B9) is term-by-term differentiable, it is sufficient to show that the differentiated series converges uniformly for z ∈ [z1, z2] (Kaplan, 1993, Sec. 6.14, Theorem 33). The proof of the uniform convergence of the differentiated series follows from the identity (Hinton, 1979)
ϕnλλn=ddzG(z,,λ),Φn=zG(z,,λ),Φn
(B16)
and a similar argument.□

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