On the Discrete Normal Modes of Quasigeostrophic Theory

Houssam Yassin; Stephen M. Griffies

doi:10.1175/JPO-D-21-0199.1

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Journal of Physical Oceanography

Abstract
1. Introduction
2. Mathematics of the quasigeostrophic phase space
3. Rossby waves in a quiescent ocean
4. Eigenfunction expansions
5. Discussion
- Galerkin approximations with L2 modes
6. Conclusions
Acknowledgments.
Data availability statement.
APPENDIX A
- Sturm–Liouville Eigenvalue Problems with λ-Dependent Boundary Conditions
APPENDIX B
- Polarization Relations and the Vertical Velocity Eigenvalue Problem
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Fig. 1.

Polar plots of the absolute value of the nondimensional angular frequency |ω_n|/(βL_d) of the first five modes of the traditional eigenvalue problem (section 3a) as a function of the wave propagation direction, k/|k|, for constant stratification. The outer most ellipse, with the largest absolute angular frequency, represents the angular frequency of the barotropic (n = 0) mode. The higher modes have smaller absolute frequencies and are thus concentric and within the barotropic angular frequency curve. Since the absolute value of the angular frequency of the barotropic mode becomes infinitely large at small horizontal wavenumbers k, we have chosen a large wavenumber k, given by kL_d = 7, so that the angular frequency of the first five modes can be plotted in the same figure. We have chosen f₀ = 10⁻⁴ s⁻¹, β = 10⁻¹¹ m⁻¹ s⁻¹, N₀ = 10⁻² s⁻¹ and H = 1 km leading to a deformation radius L_d = N₀H/f₀ = 100 km. Numerical solutions to all eigenvalue problems in this paper are obtained using Dedalus (Burns et al. 2020).

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Fig. 2.

Polar plots of the absolute value of the nondimensional angular frequency |ω_n|/(βL_d) of the first five modes from section 3b as a function of the wave propagation direction k/|k| for a horizontal wavenumber given by kL_d = 7 in constant stratification. The dashed line corresponds to ω₀; this mode becomes boundary trapped at large wavenumbers k = |k|. The remaining modes, ω_n for n = 1, 2, 3, and 4, are shown with solid lines. White regions are angles where γ₁ > 0. All Rossby waves with a propagation direction lying in the white region have negative angular frequencies ω_n and so have a westward phase speed. Gray regions are angles where γ₁ < 0. Here, ω₀ is positive while the remaining angular frequencies ω_n for n > 0 are negative. Consequently, in the gray regions, ω₀ corresponds to a Rossby wave with an eastward phase speed whereas the remaining Rossby waves have westward phase speeds. The lower boundary buoyancy gradient, proportional to ∇g₁, points toward 55^° and corresponds to a bottom slope of |∇h₁| = 1.5 × 10⁻⁵, leading to γ₁/H = 0.15. The remaining parameters are as in Fig. 1.

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Fig. 3.

The two limits of the boundary-trapped surface quasigeostrophic waves, as discussed in section 3c. (a) Convergence to the step mode given in Eq. (39) with j = 1 as γ₁ → 0⁻ for three values of γ₁ at a wavenumber k = |k| given by kL_d = 1. The phase speed approaches zero in the limit γ₁ → 0⁻. (b) Here, γ₁/H ≈ 10 for the three vertical structures G_n shown. Consequently, the bottom trapped wave has λ ≈ −k² and the phase speeds are large. The vertical structure G for three values of kL_d is shown, illustrating the dependence on k of this mode, which behaves as a boundary-trapped exponential mode with an e-folding scale of |λ|^−1/2 = k⁻¹. In both (a) and (b), the wave propagation direction θ = 260^°. All other parameters are identical to Fig. 2.

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Fig. 4.

As in Fig. 2, but now with an upper slope |∇h₂| = 10⁻⁵ in the direction 200^° in addition to the bottom slope in Fig. 2. The upper slope corresponds to γ₂/H = 0.1. The dotted line corresponds to ω₀ and the dashed line corresponds to ω₁, with these two modes becoming boundary trapped at large wavenumbers k. The remaining modes, ω_n for n = 2, 3, and 4, are shown with solid lines. White regions are angles where γ₁ > 0 and γ₂ > 0. All Rossby waves with a propagation direction lying in the white region have negative angular frequencies ω_n and so have a westward phase speed. Gray regions are angles where γ₁ < 0 and γ₂ < 0. The two gravest angular frequencies ω₀ and ω₁ are both positive while the remaining angular frequencies ω_n for n > 1 are negative. Consequently, in the gray regions, ω₀ and ω₁ each correspond to a Rossby waves with an eastward phase speed whereas the remaining Rossby waves have westward phase speeds. Stippled regions are angles where γ₁ > 0 and γ₂ < 0. In the stippled region, ω₀ is positive and has an eastward phase speed. The remaining Rossby waves in the stippled region have negative angular frequencies and have westward phase speeds.

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Fig. 5.

This figure illustrates the dependence of the vertical structure G_n of the streamfunction on the horizontal wavevector k as discussed in section 3c, for three propagation directions θ = (a),(b) 180°, (c),(d) 225°, and (e),(f) 265° [e.g., the row containing (a) and (b) are the vertical structures of waves at θ = 180°] and two wavenumbers kL_d = (left) 0.5 and (right) 7 (where k = |k|) [e.g., (b), (d), and (f) are the vertical structure of waves with kL_d = 7]. The parameters for the above figure are identical to Fig. 2. We emphasize two features in this figure. First, note how the boundary modes (n = 0, 1) are typically only boundary-trapped at small horizontal scales (i.e., for kL_d = 7). At larger horizontal scales, we typically obtain a depth-independent mode along with another mode with large-scale features in the vertical direction. Second, note that for γ₁ and γ₂ > 0, as in (a) and (b), the nth mode has n internal zeros, as in Sturm–Liouville theory; for γ₁ > 0 and γ₂ < 0, as in (c) and (d), the first two modes (n = 0, 1) have no internal zeros; and for γ₁ and γ₂ < 0, the zeroth-mode G₀ has one internal zero, the first and second modes (G₁ and G₂) have no internal zeros, and the third mode G₂ has one internal zero. The zero crossing for the n = 0 mode in (f) is difficult to observe because the amplitude of G₀ is small near the zero crossing.

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Fig. 6.

The first six vertical velocity normal modes χ_n (thin gray lines) and streamfunction normal modes G_n (black lines) (see section 3d). The propagation direction is θ = 75^° with a wavenumber of kL_d = 2. The remaining parameters are as in Fig. 2. Note that χ_n and G_n are nearly indistinguishable from the boundary-trapped modes n = 0, 1 whereas they are related by a vertical derivative for the internal modes n > 1. The eigenvalue in the figure is nondimensionalized by the deformation radius L_d.

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Article Type:: Research Article

On the Discrete Normal Modes of Quasigeostrophic Theory

Houssam Yassin

Houssam Yassin ^aProgram in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey

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and

Stephen M. Griffies

Stephen M. Griffies ^aProgram in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey
^bNOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey

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Online Publication:: 25 Feb 2022

Print Publication:: 01 Feb 2022

DOI:: https://doi.org/10.1175/JPO-D-21-0199.1

Page(s):: 243–259

Received:: 10 Sep 2021

Accepted:: 09 Nov 2021

Published Online:: 25 Feb 2022

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

The discrete baroclinic modes of quasigeostrophic theory are incomplete, and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal-mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradients activates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence of multiple modes with no internal zeros, a finite number of modes with negative norms, and the fact that their vertical structures form a basis capable of representing any quasigeostrophic state with a differentiable series expansion. These properties are a consequence of the Pontryagin-space setting of the Rossby wave problem in the presence of boundary buoyancy gradients (as opposed to the usual Hilbert-space setting). We also examine the quasigeostrophic vertical velocity modes and derive a complete basis for such modes as well. A natural application of these modes is the development of a weakly nonlinear wave-interaction theory of geostrophic turbulence that takes topography into account.

Corresponding author: Houssam Yassin, hyassin@princeton.edu

Abstract

Corresponding author: Houssam Yassin, hyassin@princeton.edu

Keywords: Rossby waves; Differential equations; Fourier analysis

1. Introduction

a. Background

The vertical decomposition of quasigeostrophic motion into normal modes plays an important role in bounded stratified geophysical fluids (e.g., Charney 1971; Flierl 1978; Fu and Flierl 1980; Wunsch 1997; Chelton et al. 1998; Smith and Vallis 2001; Tulloch and Smith 2009; Lapeyre 2009; Ferrari et al. 2010; Ferrari and Wunsch 2010; de La Lama et al. 2016; LaCasce 2017; Brink and Pedlosky 2019). Most prevalent are the traditional baroclinic modes (e.g., section 6.5.2 in Vallis 2017) that are the vertical structures of Rossby waves in a quiescent ocean with no topography or boundary buoyancy gradients. In a landmark contribution, Wunsch (1997) partitions the ocean’s kinetic energy into the baroclinic modes and finds that the zeroth and first baroclinic modes dominate over most of the extratropical ocean. Additionally, Wunsch (1997) concludes that the surface signal primarily reflects the first baroclinic mode and, therefore, the motion of the thermocline.

However, the use of baroclinic modes has come under increasing scrutiny in recent years (Lapeyre 2009; Roullet et al. 2012; Scott and Furnival 2012; Smith and Vanneste 2012). Lapeyre (2009) observes that the vertical shear of the baroclinic modes vanishes at the boundaries, thus leading to the concomitant vanishing of the boundary buoyancy. Consequently, Lapeyre (2009) proposes that the baroclinic modes cannot be complete^¹ due to their inability to represent boundary buoyancy. To supplement the baroclinic modes, Lapeyre (2009) includes a boundary-trapped exponential surface quasigeostrophic solution (see Held et al. 1995) and suggests that the surface signal primarily reflects, not thermocline motion, but boundary-trapped surface quasigeostrophic dynamics (see also Lapeyre 2017).

Appending additional functions to the collections of normal modes as in Lapeyre (2009) or Scott and Furnival (2012) does not result in a set of normal modes since the appended functions are not orthogonal to the original modes. It is only with Smith and Vanneste (2012) that a set of normal modes capable of representing arbitrary surface buoyancy is derived.

Yet it is not clear how the normal modes of Smith and Vanneste (2012) differ from the baroclinic modes or what these modes correspond to in linear theory. Indeed, Rocha et al. (2015), noting that the baroclinic series expansion of any sufficiently smooth function converges uniformly to the function itself, argues that the incompleteness of the baroclinic modes has been “overstated.” Moreover, de La Lama et al. (2016) and LaCasce (2017), motivated by the observation that the leading empirical orthogonal function of Wunsch (1997) vanishes near the ocean bottom, propose an alternate set of modes—the surface modes—that have a vanishing pressure at the bottom boundary.

We thus have a variety of proposed normal modes, and it is not clear how their properties differ. Are the baroclinic modes actually incomplete? What about the surface modes? What does completeness mean in this context? The purpose of this paper is to answer these questions.

b. Normal modes and eigenfunctions

A normal mode is a linear motion in which all components of a system move coherently at a single frequency. Mathematically, a normal mode has the form

Φ_{a} (x, y, z) e^{- i ω_{a} t},

(1)

where Φ_a describes the spatial structure of the mode and ω_a is its angular frequency. The function Φ_a is obtained by solving a differential eigenvalue problem and hence is an eigenfunction. The collection of all eigenfunctions forms a basis of some function space relevant to the problem.

By an abuse of terminology, the spatial structure, Φ_a, is often called a normal mode (e.g., the term “Fourier mode” is often used for e^ikx, where k is a wavenumber). In linear theory, this misnomer is often benign because each Φ_a corresponds to a frequency ω_a. For example, given some initial condition Ψ(x, y, z), we decompose Ψ as a sum of modes at t = 0,

Ψ (x, y, z) = \sum_{a} c_{a} Φ_{a} (x, y, z),

(2)

where the c_a are the Fourier coefficients, and the time evolution is then given by

\sum_{a} c_{a} Φ_{a} (x, y, z) e^{- i ω_{a} t} .

(3)

However, with nonlinear dynamics, this abuse of terminology can be confusing. Given some spatial structure Ψ(x, y, z) in a fluid whose flow is nonlinear, we can still exploit the basis properties of the eigenfunctions Φ_a to decompose Ψ as in Eq. (2). Whereas in a linear fluid only wave motion of the form in Eq. (1) is possible, a nonlinear flow admits a larger collection of solutions (e.g., nonlinear waves and coherent vortices) and so the linear wave solution Eq. (3) no longer follows from the decomposition Eq. (2).

For this reason, we call the linear solution in Eq. (1) a physical normal mode to distinguish it from the spatial structure Φ_a, which is only an eigenfunction. Otherwise, we will use the terms “normal mode” and “eigenfunction” interchangeably to refer to the spatial structure Φ_a, as is prevalent in the literature.

Our strategy here is then the following. We find the physical normal modes [of the form Eq. (1)] to various Rossby wave problems and examine the basis properties of their constituent eigenfunctions. By assuming a doubly periodic domain in the horizontal, the problem reduces to finding the vertical normal modes F_n(z). These vertical normal modes are obtained by solving a differential eigenvalue problem of the form

- \frac{d}{d z} (p \frac{d F}{d z}) + q F = λ r F

(4)

in the interval (z₁, z₂) with boundary conditions

- [a_{j} F - b_{j} (p \frac{d F}{d z}) (z_{j})] = λ [c_{j} F (z_{j}) - d_{j} (p \frac{d F}{d z}) (z_{j})]

(5)

for j = 1, 2, where 1/p(z), q(z), and r(z) are real-valued functions and a_j, b_j, c_j, and d_j are real numbers. This eigenvalue problem in Eqs. (4)–(5) differs from traditional Sturm–Liouville problems in that the eigenvalue λ appears in the boundary conditions Eq. (5). Our goal is to find a collection of eigenfunctions F_n (i.e., so-called normal modes in the prevalent terminology) capable of representing every possible quasigeostrophic state.

c. Contents of this article

This article constitutes an examination of all collections of discrete (i.e., noncontinuum^²) quasigeostrophic normal modes. We include the baroclinic modes, the surface modes of de La Lama et al. (2016) and LaCasce (2017), the surface-aware mode of Smith and Vanneste (2012), as well as various generalizations. To study the completeness of a set of normal modes, we must first define the underlying space in question. From general considerations, we introduce in section 2 the quasigeostrophic phase space, defined as the space of all possible quasigeostrophic states. Subsequently, in section 3 we use the general theory of differential eigenvalue problems with eigenvalue-dependent boundary conditions, as developed in Yassin (2021), to study Rossby waves in an ocean with prescribed boundary buoyancy gradients (e.g., topography; see section 2a). Intriguingly, in an ocean with no topography, we find that, in addition to the usual baroclinic modes, there are two additional stationary step-mode solutions that have not been noted before. The stationary step modes are the limits of boundary-trapped surface quasigeostrophic waves as the boundary buoyancy gradient vanishes.

Our study of Rossby waves then leads us examine all possible discrete collections of normal modes in section 4. As shown in this section, the baroclinic modes are incomplete, as argued by Lapeyre (2009), and we point out that the incompleteness leads to a loss of information after projecting a function onto the baroclinic modes. In contrast, modes such as those suggested by Smith and Vanneste (2012) are complete in the quasigeostrophic phase space so that projecting a function onto such modes provides an equivalent representation of the function.

We offer discussion of our analysis in section 5 and conclusions in section 6. Appendix A summarizes the key mathematical results pertaining to eigenvalue problems where the eigenvalue appears in the boundary conditions. Appendix B then summarizes the polarization relations as well as the vertical velocity eigenvalue problem.

2. Mathematics of the quasigeostrophic phase space

a. The potential vorticity

Consider a three-dimensional region D of the form

D = D_{0} \times (z_{1}, z_{2}) .

(6)

The area of the lower and upper boundaries is denoted by D₀ and is a rectangle of area A while z₁ (lower boundary) and z₂ (upper boundary) are constants. The horizontal boundaries are either rigid or periodic.

The state of a quasigeostrophic fluid in D is determined by a charge-like quantity known as the quasigeostrophic potential vorticity (Hoskins et al. 1985; Schneider et al. 2003). If the potential vorticity is distributed throughout the three-dimensional region D, we are concerned with the volume potential vorticity density Q, with Q related to the geostrophic streamfunction ψ by [e.g., section 5.4 of Vallis (2017)]

Q = f + \nabla^{2} ψ + \frac{\partial}{\partial z} (\frac{f_{0}^{2}}{N^{2}} \frac{\partial ψ}{\partial z}) .

(7)

Here, the latitude-dependent Coriolis parameter is

f = f_{0} + β y,

(8)

N(z) is the prescribed background buoyancy frequency, ∇² is the horizontal Laplacian operator, and

u = \hat{z} \times \nabla ψ

(9)

is the horizontal geostrophic velocity,

u = (u, υ)

Additionally, the potential vorticity may be distributed over a two-dimensional region, say the lower and upper boundaries D₀, to obtain surface potential vorticity densities R₁ and R₂. The surface potential vorticity densities are related to the streamfunction by

R_{j} = {(- 1)}^{j + 1} [g_{j} + {(\frac{f_{0}^{2}}{N^{2}} \frac{\partial ψ}{\partial z}) |}_{z = z_{j}}]

(10)

where g_j is an imposed surface potential vorticity density at the lower or upper boundary and j = 1, 2. The density g_j corresponds to a prescribed buoyancy

b_{j} = \frac{N^{2}}{f_{0}} g_{j}

(11)

at the jth boundary [see Eq. (B6)]. Alternatively, g_j may be thought of as an infinitesimal topography through

g_{j} = f_{0} h_{j}

(12)

where h_j represents infinitesimal topography at the jth boundary. Whereas Q has dimensions of inverse time, R_j has dimensions of length per time.

b. Defining the quasigeostrophic phase space

We define the quasigeostrophic phase space to be the space of all possible quasigeostrophic states, with a quasigeostrophic state determined by the potential vorticity densities Q, R₁, and R₂. Note that the volume potential vorticity density Q is defined throughout the whole fluid region D so that Q = Q(x, y, z, t). In contrast, the surface potential vorticity densities R₁ and R₂ are only defined on the two-dimensional lower and upper boundary surfaces D₀ so that R_j = R_j(x, y, t).

It is useful to restate the previous paragraph with some added mathematical precision. For that purpose, let L²[D] be the space of square-integrable functions^³ in the fluid volume D and let L²[D₀] be the space of square-integrable functions on the boundary area D₀. Elements of L²[D] are functions of three spatial coordinates, whereas elements of L²[D₀] are functions of two spatial coordinates. Hence, Q ∈ L²[D] and R₁, R₂ ∈ L²[D₀].

Define the space

P

P = L^{2} [D] \oplus L^{2} [D_{0}] \oplus L^{2} [D_{0}],

(13)

where ⊕ is the direct sum. Equation (13) states that any element of

P

is a tuple (Q, R₁, R₂) of three functions, where Q = Q(x, y, z, t) is a function on the volume D and hence element of L²[D] while the functions R_j = R_j(x, y, t), for j = 1, 2, are functions on the area D₀ and hence are elements of L²[D₀]. We conclude that (Q, R₁, R₂) ∈

P

and that

P

is the space of all possible quasigeostrophic states. We thus call

P

the quasigeostrophic phase space.

c. The phase space in terms of the streamfunction

Given an element (Q, R₁, R₂) ∈

P

, we can reconstruct a continuous function ψ that contains the same dynamical information as (Q, R₁, R₂). By inverting the problem

Q - f = \nabla^{2} ψ_{int} + \frac{\partial}{\partial z} (\frac{f_{0}}{N^{2}} \frac{\partial ψ_{int}}{\partial z}) for z \in (z_{1}, z_{2}),

R_{1} - g_{1} = \frac{f_{0}^{2}}{N^{2}} \frac{\partial ψ_{low}}{\partial z} for z = z_{1}

R_{2} + g_{2} = - \frac{f_{0}^{2}}{N^{2}} \frac{\partial ψ_{upp}}{\partial z} for z = z_{2},

(14)

we obtain a function ψ(x, y, z) that is unique up to a gauge transformation (see Schneider et al. 2003). Conversely, given a function ψ(x, y, z), we can differentiate ψ as in Eqs. (14) to obtain (Q, R₁, R₂) ∈

P

. Thus, we can also consider the quasigeostrophic phase space

P

to be the space of all possible streamfunctions ψ.

Equations (14) motivate the definition of the relative potential vorticity densities, q = Q − f and r_j = R_j − (−1)^j⁺¹g_j, which are the portions of the potential vorticity providing a source for a streamfunction. Explicitly, the relative potential vorticity densities are

q = \nabla^{2} ψ + \frac{\partial}{\partial z} (\frac{f_{0}^{2}}{N^{2}} \frac{\partial ψ}{\partial z}) for z \in (z_{1}, z_{2}),

(15a)

r_{1} = \frac{f_{0}^{2}}{N^{2}} \frac{\partial ψ}{\partial z} for z = z_{1}, and

(15b)

r_{2} = - \frac{f_{0}^{2}}{N^{2}} \frac{\partial ψ}{\partial z} for z = z_{2} .

(15c)

d. The vertical structure phase space

We proceed by expanding the potential vorticity density distribution, (q, r₁, r₂), and the streamfunction ψ in terms of the eigenfunctions e_k of the horizontal Laplacian. For the rectangular horizontal domain D₀, the eigenfunction e_k(x) satisfies

- \nabla^{2} e_{k} = k^{2} e_{k} .

(16)

where x = (x, y) is the horizontal position vector, k = (k_x, k_y) is the horizontal wavevector, and k = |k| is the horizontal wavenumber. For example, in a horizontally periodic domain the eigenfunctions e_k(x) are proportional to complex exponentials eⁱ^k^⋅^x.

Projecting the relative potential vorticity density distribution, (q, r₁, r₂), onto the horizontal eigenfunctions e_k yields

q (x, z, t) = \sum_{k} q_{k} (z, t) e_{k} (x), for z \in (z_{1}, z_{2}), and

(17a)

r_{j} (x, t) = \sum_{k} r_{j k} (t) e_{k} (x) for j = 1, 2.

(17b)

Thus the Fourier coefficients of (q, r₁, r₂) are (q_k, r₁_k, r₂_k) where q_k is a function of z, and r₁_k and r₂_k are independent of z. Hence, q_k is an element of L²[(z₁, z₂)] whereas r₁_k and r₂_k are elements of the space of complex numbers^⁴ ℂ.

We conclude that the vertical structure of the potential vorticity, given by (q_k, r₁_k, r₂_k), is an element of

\hat{P} = L^{2} [(z_{1}, z_{2})] \oplus ℂ \oplus ℂ,

(18)

so that the vertical structures of the potential vorticity distribution are determined by a function q_k in L²[(z₁, z₂)] and two z-independent elements r₁_k and r₂_k of ℂ. Similarly, the streamfunction can be represented as

ψ (x, z, t) = \sum_{k} ψ_{k} (z, t) e_{k} (x),

(19)

where ψ_k and (q_k, r₁_k, r₂_k) are related by

q_{k} = - k^{2} ψ_{k} + \frac{\partial}{\partial z} (\frac{f_{0}^{2}}{N^{2}} \frac{\partial ψ_{k}}{\partial z}) and

(20a)

r_{j k} = {(- 1)}^{j + 1} {(\frac{f_{0}^{2}}{N^{2}} \frac{\partial ψ_{k}}{\partial z}) |}_{z = z_{j}} .

(20b)

As before, knowledge of the vertical structure of the streamfunction ψ_k(z) is equivalent to knowing the vertical structure of the potential vorticity distribution (q_k, r_1k, r_2k). In the resulting differential eigenvalue problem for the vertical normal modes, the nonzero r_j_k lead to an eigenvalue problem of the form given in Eqs. (4) and (5), with the eigenvalue appearing in the boundary condition. Such an eigenvalue problem takes place in the space $\hat{P}$ given by Eq. (18) (Yassin 2021). Thus $\hat{P}$ is also the space of all possible streamfunction vertical structures.

That ψ_k belongs to $\hat{P}$ and not L²[(z₁, z₂)] underlies much of the confusion over baroclinic modes. Assertions of completeness, based on Sturm–Liouville theory, assume that ψ is an element of L²[(z₁, z₂)]. However, as we have shown, that is an incorrect assumption. That ψ belongs to $\hat{P}$ will have consequences for the convergence and differentiability of normal-mode expansions, as discussed in section 4. In the context of quasigeostrophic theory, the space $\hat{P}$ first appeared in Smith and Vanneste (2012). More generally, $\hat{P}$ appears in the presence of nontrivial boundary dynamics (Yassin 2021).

We call

\hat{P}

the vertical structure phase space, and for convenience we denote L²[(z₁, z₂)] by L² for the remainder of the article. The vertical structure phase space

\hat{P}

is then written as the direct sum

\hat{P} = L^{2} \oplus ℂ^{2} .

(21)

e. Representing the energy and potential enstrophy

We find it convenient to represent several quadratic quantities in terms of the eigenfunctions of the horizontal Laplacian e_k(x). The energy per unit mass in the volume D is given by

E = \frac{1}{V} \int_{D} [{| \nabla ψ |}^{2} + \frac{f_{0}^{2}}{N^{2}} {| \frac{\partial ψ}{\partial z} |}^{2}] d A d z = \sum_{k} E_{k},

(22)

where the horizontal energy mode is given by the vertical integral

E_{k} = \frac{1}{H} \int_{z_{1}}^{z_{2}} [k^{2} {| ψ_{k} |}^{2} + \frac{f_{0}^{2}}{N^{2}} {| \frac{\partial ψ_{k}}{\partial z} |}^{2}] d z,

(23)

with V = AH being the domain volume and H = z₂ − z₁ being the domain depth.

Similarly, for the relative volume potential enstrophy density Z, we have

Z = \frac{1}{V} \int_{D} {| q |}^{2} d A d z = \sum_{k} Z_{k}

(24)

where

Z_{k} = \frac{1}{H} \int_{z_{1}}^{z_{2}} {| q_{k} |}^{2} d z .

(25)

Last, analogous to Z, we have the relative surface potential enstrophy densities Y_j on the area D₀:

Y_{j} = \frac{1}{A} \int_{D_{0}} {| r_{j} |}^{2} d A = \sum_{k} Y_{j k}

(26)

where

Y_{j k} = {| r_{j k} |}^{2} .

(27)

3. Rossby waves in a quiescent ocean

In this section, we study Rossby waves in an otherwise quiescent ocean; in other words, we examine the physical normal modes of a quiescent ocean. The linear equations of motion are

\frac{\partial q}{\partial t} + β υ = 0 for z \in (z_{1}, z_{2}) and

(28a)

\frac{\partial r_{j}}{\partial t} + u \cdot \nabla [{(- 1)}^{j + 1} g_{j}] = 0 for z = z_{j} .

(28b)

We assume that the prescribed surface potential vorticity densities at the lower and upper boundaries, g₁ and g₂, are linear, which ensures that the resulting eigenvalue problem is separable. Moreover, because the ocean is quiescent, g₁ and g₂ must refer to topographic slopes, as in Eq. (12).

The importance of the linear problem in Eq. (28) is that it provides all possible discrete Rossby wave normal modes in a quasigeostrophic flow. Substituting a wave ansatz of the form [cf. Eq. (1) for physical normal modes]

ψ (x, z, t) = \hat{ψ} (z) e_{k} (x) e^{- i ω t}

(29)

into the linear problem Eq. (28) renders

(- i ω) [- k^{2} \hat{ψ} + \frac{d}{d z} (\frac{f_{0}^{2}}{N^{2}} \frac{d \hat{ψ}}{d z})] + i k_{x} β \hat{ψ} = 0

(30)

for z ∈ (z₁, z₂), and

(- i ω) (\frac{f_{0}^{2}}{N^{2}} \frac{d \hat{ψ}}{d z}) + i \hat{z} \cdot (k \times \nabla g_{j}) \hat{ψ} = 0

(31)

for z = z₁, z₂.

a. Traditional Rossby wave problem

We first examine the traditional case of linear fluctuations to a quiescent ocean with isentropic lower and upper boundaries, that is, with no topography. Setting ∇g₁ = ∇g₂ = 0 in the eigenvalue problem of Eqs. (30)–(31) gives

ω [- k^{2} F + \frac{d}{d z} (\frac{f_{0}^{2}}{N^{2}} \frac{d F}{d z})] - β k_{x} F = 0 and

(32a)

{ω (\frac{f_{0}^{2}}{N^{2}} \frac{d F}{d z}) |}_{z = z_{j}} = 0,

(32b)

where

\hat{ψ} (z) = {\hat{ψ}}_{0} F (z)

and F is a nondimensional function. There are two cases to consider depending on whether ω vanishes.

1) Traditional baroclinic modes

Assuming ω ≠ 0 in the eigenvalue problem Eq. (32) renders a Sturm–Liouville eigenvalue problem in L²:

- \frac{d}{d z} (\frac{f_{0}^{2}}{N^{2}} \frac{d F}{d z}) = λ F for z \in (z_{1}, z_{2}) and

(33a)

\frac{f_{0}^{2}}{N^{2}} \frac{d F}{d z} = 0 for z = z_{1}, z_{2},

(33b)

where the eigenvalue λ is given by

λ = - k^{2} - \frac{β k_{x}}{ω} .

(34)

See Fig. 1 for an illustration of the dependence of |ω| on the wavevector k.

Fig. 1.

Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPO-D-21-0199.1

Fig. 2.

Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPO-D-21-0199.1

From Sturm–Liouville theory (e.g., Brown and Churchill 1993), the eigenvalue problem Eq. (33) has infinitely many eigenfunctions, F₀, F₁, F₂, … with distinct and ordered eigenvalues λ_n satisfying

0 = λ_{0} < λ_{1} < \dots \to \infty .

(35)

The nth mode, F_n, has n internal zeros in the interval (z₁, z₂). The eigenfunctions are orthonormal with respect to the inner product [,], given by the vertical integral

[F, G] = \frac{1}{H} \int_{z_{1}}^{z_{2}} F G d z,

(36)

with orthonormality, meaning that

δ_{m n} = [F_{m}, F_{n}],

(37)

where δ_mn is the Kronecker delta. A powerful and commonly used result of Sturm–Liouville theory is that the set

{F_{n}}_{n = 0}^{\infty}

forms an orthonormal basis of L².

2) Stationary step modes

There are two additional solutions to the Rossby wave eigenvalue problem Eq. (32) that have not previously been noted in the literature. If ω = 0, then the eigenvalue problem in Eq. (32) becomes

β k_{x} F = 0 for z \in (z_{1}, z_{2}) and

(38a)

0 = 0 for z = z_{1}, z_{2} .

(38b)

Consequently, if k_x ≠ 0, then F(z) = 0 for z ∈ (z₁, z₂). That is, F must vanish in the interior of the interval. However, since ω = 0 in Eq. (32b), we obtain tautological boundary conditions Eq. (38b). As a result, F can take arbitrary values at the lower and upper boundaries. Thus, two solutions are

F_{j}^{step} (z) = {\begin{array}{l} 1 for z = z_{j} \\ 0 otherwise \end{array} .

(39)

The two step-mode solutions in Eq. (39) are independent of the traditional baroclinic modes F_n(z). An expansion of the step mode

F_{j}^{step}

in terms of the baroclinic modes will fail and produce a series that is identically zero.

The two stationary step modes, $F_{1}^{step}$ and $F_{2}^{step}$ , correspond to the two inert degrees of freedom in the eigenvalue problem Eq. (32). These two solutions are neglected in the traditional eigenvalue problem Eq. (33) through the assumption that ω ≠ 0. Although dynamically trivial, we will see that these two step waves are obtained as limits of boundary-trapped modes as the boundary buoyancy gradients N²∇g_j/f₀ become small.

3) The general solution

For a wavevector k with k_x ≠ 0, the vertical structure of the streamfunction must be of the form

Ψ (z) + \sum_{j = 1}^{2} Ψ_{j} F_{j}^{step} (z) = ψ_{k} (z, t = 0),

(40)

where Ψ(z) is a twice differentiable function satisfying dΨ(z_j)/dz = 0 for j = 1, 2 and Ψ₁ and Ψ₂ are arbitrary constants. We can represent Ψ according to the expansion

Ψ = \sum_{n = 0}^{\infty} [Ψ, F_{n}] F_{n},

(41)

and so the time evolution is

ψ_{k} (z, t) = \sum_{n = 0}^{\infty} [Ψ, F_{n}] F_{n} e^{- i ω_{n} t} + \sum_{j = 1}^{2} Ψ_{j} F_{j}^{step} .

(42)

It is this time-evolution expression, which is valid only in linear theory for a quiescent ocean, that gives the baroclinic modes a clear physical meaning. More precisely, Eq. (42) states that the vertical structure Ψ(z) disperses into its constituent Rossby waves with vertical structures F_n. Outside the linear theory of this section, baroclinic modes do not have a physical interpretation, although they remain a mathematical basis for L².

b. The Rhines problem

We now examine the case with a sloping lower boundary, ∇g₁ ≠ 0, and an isentropic upper boundary, ∇g₁ = 0. The special case of a meridional bottom slope and constant stratification was first investigated by Rhines (1970). Subsequently, Charney and Flierl (1981) extended the analysis to realistic stratification and Straub (1994) examined the dependence of the waves on the propagation direction. Yassin (2021) applies the mathematical theory of eigenvalue problems with λ-dependent boundary conditions and obtains various completeness and expansion results as well as a qualitative theory for the streamfunction modes. Below, we generalize these results, study the two limiting boundary conditions, and consider the corresponding vertical velocity modes.

1) The eigenvalue problem

Let

\hat{ψ} (z) = {\hat{ψ}}_{0} G (z)

, where G is a nondimensional function. We then manipulate the eigenvalue problem in Eqs. (30)–(31) to obtain (assuming ω ≠ 0)

- \frac{d}{d z} (\frac{f_{0}^{2}}{N^{2}} \frac{d G}{d z}) = λ G for z \in (z_{1}, z_{2})

(43a)

- k^{2} G - γ_{1}^{- 1} (\frac{f_{0}^{2}}{N^{2}} \frac{d G}{d z}) = λ G for z = z_{1}, and

(43b)

\frac{d G}{d z} = 0 for z = z_{2},

(43c)

where the length scale γ_j is given by

γ_{j} = {(- 1)}^{j + 1} \frac{\hat{z} \cdot (k \times \nabla g_{j})}{\hat{z} \cdot (k \times \nabla f)} = {(- 1)}^{j + 1} (\frac{α_{j} k}{β k_{x}}) \sin (Δ θ_{j}),

(44)

where α_j = |∇g_j| and Δθ_j is the angle between the wavevector k and ∇g_j measured counterclockwise from k. The parameter γ_j depends only on the direction of the wavevector k and not its magnitude k = |k|. If γ_j = 0, then the jth boundary condition can be written as a λ-independent boundary condition [as in the upper boundary condition at z = z₂ of the eigenvalue problem Eq. (43)]. For now, we assume that γ₁ ≠ 0.

Since the eigenvalue λ appears in the differential equation and one boundary condition in the eigenvalue problem Eq. (43), the eigenvalue problem takes place in L² ⊕ ℂ.

2) Characterizing the eigensolutions

The following is obtained by applying the theory summarized in appendix A to the eigenvalue problem Eq. (43).^⁵

The eigenvalue problem Eq. (43) has a countable infinity of eigenfunctions G₀, G₁, G₂, … with ordered and distinct nonzero eigenvalues λ_n satisfying

λ_{0} < λ_{1} < λ_{2} < \dots \to \infty .

(45)

The inner product 〈,〉 induced by the eigenvalue problem Eq. (43) is

〈 F, G 〉 = \frac{1}{H} [\int_{z_{1}}^{z_{2}} F G d z + γ_{1} F (z_{1}) G (z_{1})],

(46)

which depends on the direction of the horizontal wavevector k through γ₁. Moreover, γ₁ is not necessarily positive,^⁶ with one consequence being that some functions G may have a negative square: 〈G, G〉 < 0. Orthonormality of the modes G_n then takes the form

\pm δ_{m n} = 〈 G_{m}, G_{n} 〉,

(47)

where at most one mode, G_n, satisfies 〈G_n, G_n〉 = −1. The eigenfunctions

{G_{n}}_{n = 0}^{\infty}

form an orthonormal basis of L² ⊕ ℂ under the inner product Eq. (46).

Appendix A provides the following inequality,

(k^{2} + λ_{n}) 〈 G_{n}, G_{n} 〉 > 0,

(48)

that, using the dispersion relation in Eq. (34), implies that modes G_n with 〈G_n, G_n〉 > 0 correspond to waves with a westward phase speed whereas modes G_n with 〈G_n, G_n〉 < 0 correspond to waves with an eastward phase speed (assuming β > 0).

We distinguish the following cases depending on the sign of γ₁. In the following, we assume k ≠ 0.

γ₁ > 0: All eigenvalues satisfy λ_n > −k², all modes satisfy 〈G_n, G_n〉 > 0, and all waves propagate westward. The nth mode G_n has n internal zeros (Binding et al. 1994). See the regions in white in Fig. 2.
γ₁ < 0: There is one mode, G₀, with a negative square, 〈G_n, G_n〉 < 0, corresponding to an eastward-propagating wave. The eastward-propagating wave nevertheless travels pseudowestward (to the left of the upslope direction for f₀ > 0). The associated eigenvalue λ₀ satisfies λ_n < −k². The remaining modes, G_n for n > 1, have positive squares, 〈G_n, G_n〉 > 0, corresponding to westward-propagating waves and have eigenvalues λ_n satisfying λ_n > −k². Both G₀ and G₁ have no internal zeros, whereas the remaining modes, G_n, have n − 1 internal zeros for n > 1 (Binding et al. 1994). See the stippled regions in Fig. 2.

To elucidate the meaning of λ_n < −k², note that a pure surface quasigeostrophic mode^⁷ has λ = −k². Thus, λ₀ < −k² means that the bottom-trapped mode decays away from the boundary more rapidly than a pure surface quasigeostrophic wave. Indeed, the limit of λ₀ → −∞ yields the bottom step mode (39) of section 3a(2).

The step-mode limit is obtained as γ₁ → 0⁻. This limit is found as either |∇g₁| → 0 for propagation directions in which γ₁ < 0 or as k becomes parallel or antiparallel to ∇g₁ (whichever limit satisfies γ₁ → 0⁻). In this limit, we obtain a step mode exactly confined at the boundary (i.e., |λ|^−1/2 = 0) with zero phase speed (see Fig. 3a). The remaining modes then satisfy the isentropic boundary condition

{(\frac{f_{0}^{2}}{N^{2}} \frac{d G_{n}}{d z}) |}_{z = z_{1}} = 0.

(49)

Fig. 3.

Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPO-D-21-0199.1

The other limit is that of |γ₁| → ∞ that is obtained as the buoyancy gradient becomes large, |∇g₁| → ∞. In this limit, the eigenvalue λ₀ → −k² (see Fig. 3b). Moreover, the phase speed of the bottom-trapped wave becomes infinite, which is an indication that the quasigeostrophic approximation breaks down. Indeed, the large buoyancy gradient limit corresponds to steep topographic slopes and so we obtain the topographically trapped internal gravity wave of Rhines (1970), which has an infinite phase speed in quasigeostrophic theory. The remaining modes then satisfy the vanishing pressure boundary condition

G (z_{1}) = 0

(50)

as in the surface modes of de La Lama et al. (2016) and LaCasce (2017).

3) The general time-dependent solution

At some wavevector k, the observed vertical structure now has the form

Ψ (z) = ψ_{k} (z, t = 0)

(51)

where Ψ is a twice continuously differentiable function satisfying dΨ(z₂)/dz = 0. For such functions we can write (see appendix A)

Ψ = \sum_{n = 0}^{\infty} \frac{〈 Ψ, G_{n} 〉}{〈 G_{n}, G_{n} 〉} G_{n} .

(52)

so that the time evolution is

ψ_{k} (z, t) = \sum_{n = 0}^{\infty} \frac{〈 Ψ, G_{n} 〉}{〈 G_{n}, G_{n} 〉} G_{n} (z) e^{- i ω_{n} t} .

(53)

Again, it is the above expression, which is valid only in linear theory with a quiescent background state, that gives the generalized Rhines modes G_n physical meaning. Outside the linear theory of this section, the generalized Rhines modes do not have any physical interpretation and instead merely serve as a mathematical basis for L² ⊕ ℂ.

Recall from section 3a that an expansion of a step mode Eq. (39) in terms of the baroclinic modes

{F_{n}}_{n = 0}^{\infty}

produces a series that is identically zero. It follows that the step modes are independent of the baroclinic modes—they constitute independent degrees of freedom. However, with the inclusion of bottom boundary dynamics, we may now expand the bottom step mode,

F_{1}^{step} (z)

, in terms of the L² ⊕ ℂ¹ modes,

{G_{n}}_{n = 0}^{\infty}

, with the expansion given by

F_{1}^{step} (z) = \frac{γ_{1}}{H} \sum_{n = 0}^{\infty} \frac{G_{n} (z_{1})}{〈 G_{n}, G_{n} 〉} G_{n} (z) .

(54)

c. The generalized Rhines problem

The general problem with topography at both the upper and lower boundaries is

- \frac{d}{d z} (\frac{f_{0}^{2}}{N^{2}} \frac{d G}{d z}) = λ G for z \in (z_{1}, z_{2})

(55a)

- k^{2} G + {(- 1)}^{j} γ_{j}^{- 1} (\frac{f_{0}^{2}}{N^{2}} \frac{d G}{d z}) = λ G for z = z_{j},

(55b)

for j = 1, 2, where the length scale γ_j is given by Eq. (44). As the eigenvalue λ appears in both boundary conditions, the eigenvalue problem in Eq. (55) takes place in L² ⊕ ℂ². The inner product now has the form

〈 F, G 〉 = \frac{1}{H} [\int_{z_{1}}^{z_{2}} F G d z + \sum_{j = 1}^{2} γ_{j} F (z_{j}) G (z_{j})],

(56)

which reduces to Eq. (46) when γ₂ = 0. Under this inner product, the eigenfunctions

{G_{n}}_{n = 0}^{\infty}

form a basis of L² ⊕ ℂ².

There are now three cases depending on the signs of γ₁ and γ₂ and as depicted in Figs. 4 and 5. In the following, we assume k ≠ 0.

Fig. 4.

Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPO-D-21-0199.1

Fig. 5.

Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPO-D-21-0199.1

γ₁ > 0 and γ₂ > 0: This corresponds to case i in section 3b. See the regions in white in Fig. 4 and Figs. 5a and 5b.
γ₁γ₂ < 0: This corresponds to case ii in section 3b. See the stippled regions in Fig. 4 and Figs. 5c and 5d.
γ₁ < 0 and γ₂ < 0: There are two modes G₀ and G₁ with negative squares, 〈G_n, G_n〉 < 0, that propagate eastward and have eigenvalues G_n satisfying G_n < −k² for n = 1, 2. The remaining modes G_n for n > 1 have positive squares, 〈G_n, G_n〉 > 0, propagate westward, and have eigenvalues, λ_n, satisfying λ_n > −k². The zeroth mode G₀ has one internal zero, the first and second modes, G₁ and G₂, have no internal zeros, and the remaining modes G_n have n − 2 internal zeros for n > 2 (Binding and Browne 1999). See the shaded regions in Figs. 2 and 4 and Figs. 5e and 5f.

d. The vertical velocity eigenvalue problem

Let

\hat{w} (z) = {\hat{w}}_{0} χ (z)

, where χ(z) is a nondimensional function. For the Rossby waves with isentropic boundaries of section 3a (the traditional baroclinic modes), the corresponding vertical velocity modes satisfy

- \frac{d^{2} χ}{d z^{2}} = λ (\frac{N^{2}}{f_{0}^{2}}) χ

(57)

with vanishing vertical velocity boundary conditions

χ (z_{j}) = 0

(58)

(see appendix B for details). The resulting modes

{χ_{n}}_{n = 0}^{\infty}

form an orthonormal basis of L² with orthonormality given by

δ_{m n} = \frac{1}{H} \int_{z_{1}}^{z_{2}} χ_{m} χ_{n} (\frac{N^{2}}{f_{0}^{2}}) d z .

(59)

One can obtain the eigenfunctions χ_n by solving the eigenvalue problem in Eqs. (57)–(58) or by differentiating the streamfunction modes F_n according to Eq. (B8).

Quasigeostrophic boundary dynamics

As seen earlier, boundary buoyancy gradients activate boundary dynamics in the quasigeostrophic problem. In this case, boundary conditions for the quasigeostrophic vertical velocity problem Eq. (57) become

- {(- 1)}^{j} γ_{j} k^{2} {\frac{d χ}{d z} |}_{z_{j}} = λ [{χ |}_{z_{j}} + {(- 1)}^{j} γ_{j} {\frac{d χ}{d z} |}_{z_{j}}]

(60)

(see appendix B). The resulting modes

{χ_{n}}_{n = 0}^{\infty}

(Fig. 6) satisfy a peculiar orthogonality relation given by Eq. (B14).

Fig. 6.

Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPO-D-21-0199.1

4. Eigenfunction expansions

Motivated by the Rossby waves of the previous section, we now investigate various sets of normal modes for quasigeostrophic theory. Let

{F_{n}}_{n = 0}^{\infty}

be a collection of continuous functions that form a basis of L² and assume ψ_k(z, t) is twice continuously differentiable in z. Define the eigenfunction expansion

ψ_{k}^{exp}

of ψ by

ψ_{k}^{exp} (z, t) = \sum_{n = 0}^{\infty} ψ_{k n} (t) F_{n} (z),

(61)

where

ψ_{k n} = [ψ_{k}, F_{n}] .

(62)

Because

{F_{n}}_{n = 0}^{\infty}

is a basis of L², the eigenfunction expansion

ψ_{k}^{exp}

satisfies (e.g., Brown and Churchill 1993)

\int_{z_{1}}^{z_{2}} {| ψ_{k} (z) - ψ_{k}^{exp} (z) |}^{2} d z = 0.

(63)

Significantly, the vanishing of the integral in Eq. (63) does not imply

ψ_{k} (z) = ψ_{k}^{exp} (z)

at every z ∈ [z₁, z₂] because the two functions can still differ at some points z ∈ [z₁, z₂].

In the following, we will only consider eigenfunction expansions that diagonalize the energy and potential enstrophy integrals of section 2e.

a. The four possible L² modes

There are only four L² bases in quasigeostrophic theory that diagonalize the energy and potential enstrophy integrals. All four sets of corresponding normal modes satisfy the differential equation

- \frac{d}{d z} (\frac{f_{0}^{2}}{N^{2}} \frac{d F}{d z}) = λ F z \in (z_{1}, z_{2}),

(64)

but differ in boundary conditions according to the following (recall that z₁ is the bottom and z₂ is the surface)—

baroclinic modes: these have vanishing vertical velocity at both boundaries (Neumann),

\frac{d F (z_{1})}{d z} = 0 and \frac{d F (z_{2})}{d z} = 0,

(65)

antibaroclinic modes: these have vanishing pressure^⁸ at both boundaries (Dirichlet),

F (z_{1}) = 0 and F (z_{2}) = 0,

(66)

surface modes: (mixed Neumann/Dirichlet)

F (z_{1}) = 0 and \frac{d F (z_{2})}{d z} = 0, and

(67)

antisurface modes: (mixed Neumann/Dirichlet)

\frac{d F (z_{1})}{d z} = 0 and F (z_{2}) = 0 .

(68)

All four sets of modes are missing two modes. Each boundary condition of the form

\frac{d F (z_{j})}{d z} = 0

(69)

implies a missing step mode, and a boundary condition of the form

F (z_{j}) = 0

(70)

implies a missing boundary-trapped exponential mode [see the γ₁ → ∞ limit leading to Eq. (50)].

b. Expansions with L² modes

We here examine the pointwise convergence and the term-by-term differentiability of eigenfunction expansions in terms of L² modes. These properties of L² Sturm–Liouville expansions may be found in Brown and Churchill (1993) and Levitan and Sargsjan (1975).^⁹

1) Pointwise equality on [z₁, z₂]

For all four sets of L² modes, if ψ_k is twice continuously differentiable in z, we obtain pointwise equality in the interior

ψ_{k} (z) = ψ_{k}^{exp} (z) for z \in (z_{1}, z_{z}) .

(71)

The behavior at the boundaries depends on the boundary conditions that the modes F_n satisfy. If the F_n satisfy the vanishing pressure boundary condition at the jth boundary

F_{n} (z_{j}) = 0,

(72)

then

ψ_{k}^{exp} (z_{j}) = 0

(73)

regardless of the values of ψ_k(z_j). It follows that

ψ_{k}^{exp}

will be continuous over (z₁, z₂) and will generally have a jump discontinuity at the boundaries [unless ψ_k(z_j) = 0 for j = 1, 2]. In contrast, if the F_n satisfy a zero vertical velocity boundary condition at the jth boundary

\frac{d F_{n} (z_{j})}{d z} = 0,

(74)

then

ψ_{k} (z_{j}) = ψ_{k}^{exp} (z_{j}) .

(75)

Consequently, of the four sets of L² modes, only with the baroclinic modes do we obtain the pointwise equality

ψ_{k} (z) = ψ_{k}^{exp} (z)

on the closed interval [z₁, z₂].

However, even though $ψ_{k}^{exp}$ converges pointwise to ψ_k when the baroclinic modes are used, we are unable to represent the corresponding velocity w_k in terms of the vertical velocity baroclinic modes since the modes vanish at both boundaries. Analogous considerations show that only the antibaroclinic vertical velocity modes (see appendix B) can represent arbitrary vertical velocities.

2) Differentiability of the series expansion

Although we obtain pointwise equality on the whole interval [z₁, z₂] with the streamfunction baroclinic modes, we have lost two degrees of freedom in the expansion process. Recall that the degrees of freedom in the quasigeostrophic phase space are determined by the potential vorticity. The volume potential vorticity q_k is associated with the L² degrees of freedom while the surface potential vorticities, r₁_k and r₂_k, are associated with the ℂ² degrees of freedom.

The series expansion

ψ_{k}^{exp}

of ψ_k in terms of the baroclinic modes is differentiable in the interior (z₁, z₂). Consequently, we can differentiate the

ψ_{k}^{exp}

series for z ∈ (z₁, z₂) to recover q_k; that is,

q_{k} = \sum_{n = 0}^{\infty} q_{k n} F_{n}

(76)

where

q_{k n} = - (k^{2} + λ_{n}) ψ_{k n} .

(77)

However,

ψ_{k}^{exp}

is not differentiable at the boundaries z = z₁, z₂, so we are unable to recover the surface potential vorticities r₁_k and r₂_k. Two degrees of freedom are lost by projecting onto the baroclinic modes.^¹⁰

The energy at wavevector k is indeed partitioned between the modes

E_{k} = \sum_{n = 0}^{\infty} (k^{2} + λ_{n}) {| ψ_{k n} |}^{2},

(78)

and similarly for the potential enstrophy

Z_{k} = \sum_{n = 0}^{\infty} {(k^{2} + λ_{n})}^{2} {| ψ_{k n} |}^{2} .

(79)

However, because we have lost r₁_k and r₂_k in the projection process, the surface potential enstrophies Y₁_k and Y₂_k, defined in Eq. (27), are not partitioned.

c. Quasigeostrophic L² ⊕ ℂ² modes

Consider the eigenvalue problem

- \frac{d}{d z} (\frac{f_{0}^{2}}{N^{2}} \frac{d G}{d z}) = λ G for z \in (z_{1}, z_{2}) and

(80a)

- k^{2} G + {(- 1)}^{j} D_{j}^{- 1} (\frac{f_{0}^{2}}{N^{2}} \frac{d G}{d z}) = λ G for z = z_{j},

(80b)

where D₁ and D₂ are nonzero real constants. This eigenvalue problem differs from the generalized Rhines eigenvalue problem in Eq. (55) in that D_j are generally not equal to the γ_j defined in Eq. (44). The inner product 〈,〉 induced by the eigenvalue problem in Eq. (80) is given by Eq. (56) with the γ_j replaced by the D_j.

Smith and Vanneste (2012) investigate an equivalent eigenvalue problem to (80) and conclude that, when D₁ and D₂ are positive, the resulting eigenfunctions form a basis of L² ⊕ ℂ². However, such a completeness result is insufficient for the Rossby wave problem of section 3c, in which case D_j = γ_j and γ_j can be negative.

d. Expansion with L² ⊕ ℂ² modes

When D₁ and D₂ in the eigenvalue problem Eq. (80) are finite and nonzero, the resulting eigenmodes

{G_{n}}_{n = 0}^{\infty}

form a basis for the vertical structure phase space L² ⊕ ℂ². Thus, the projection

ψ_{k}^{exp} (z) = \sum_{n = 0}^{\infty} ψ_{k n} G_{n} (z),

(81)

where

ψ_{k n} = \frac{〈 ψ_{k}, G_{n} 〉}{〈 G_{n}, G_{n} 〉}

(82)

is an equivalent representation of ψ_k. Not only do we have pointwise equality

ψ_{k} (z) = ψ_{k}^{exp} (z) for z \in [z_{1}, z_{2}],

(83)

but the series

ψ_{k}^{exp}

is also differentiable on the closed interval [z₁, z₂] [the case of D_j > 0 is due to Fulton (1977) whereas the case of D_j < 0 is due to Yassin (2021)]. Thus, given

ψ_{k}^{exp}

, we can differentiate to obtain both q_k and r_j_k and thereby recover all quasigeostrophic degrees of freedom. Indeed, we have

q_{k} (z, t) = \sum_{n = 0}^{\infty} q_{k n} (t) G_{n} (z) and

(84)

r_{j k} (t) = \sum_{n = 0}^{\infty} r_{j k n} (t) G_{n} (z_{j})

(85)

where

q_{k n} = - (k^{2} + λ_{n}) \frac{〈 Ψ, G_{n} 〉}{〈 G_{n}, G_{n} 〉} and

(86)

r_{j k n} = D_{j} q_{k n}

(87)

for j = 1, 2.

In addition, the energy E_k, volume potential enstrophy Z_k, and surface potential enstrophies Y₁_k and Y₂_k are partitioned (diagonalized) between the modes

E_{k} = \sum_{n = 0}^{\infty} 〈 G_{n}, G_{n} 〉 (k^{2} + λ_{n}) {| ψ_{k n} |}^{2} and

(88)

Z_{k} + \frac{1}{H} \sum_{j = 1}^{2} \frac{1}{D_{j}} Y_{j k} = \sum_{n = 0}^{\infty} 〈 G_{n}, G_{n} 〉 {(k^{2} + λ_{n})}^{2} {| ψ_{k n} |}^{2} .

(89)

5. Discussion

The traditional baroclinic modes are useful because they are the vertical structures of linear Rossby waves in a resting ocean and they can be used for wave-turbulence studies such as in, e.g., Hua and Haidvogel (1986) and Smith and Vallis (2001). Therefore, any basis we choose should not only be complete in L² ⊕ ℂ² but should also represent the vertical structure of Rossby waves in the linear (quiescent ocean) limit. Such a basis would then be amenable to wave-turbulence arguments and can permit a dynamical interpretation of field observations. The basis suggested by Smith and Vanneste (2012) does not correspond to Rossby waves in the linear limit. It is a mathematical basis with two independent parameters D₁, D₂ > 0 that diagonalizes the energy and potential enstrophy integrals.

This observation can be verified by rewriting the linear time-evolution Eqs. (28a) and (28b) as a time-evolution equation for modal amplitudes. Expanding ψ in terms of the generalized Rhines modes [the vertical eigenfunctions of the eigenvalue problem Eq. (55) with D_j = γ_j(k) for each wavevector k], using the relationship between q_k_n and r_j_k_n given by Eq. (87), the orthonormality condition Eq. (56), and assuming γ_j(k) ≠ 0, we obtain

\frac{d q_{k n}}{d t} + i β k_{x} ψ_{k n} = 0.

(90)

This expansion diagonalizes the linear terms (i.e., β and ∇g_j) in the time-evolution Eqs. (28a) and (28b) through the choice D_j = γ_j(k) for each vector k. However, this choice cannot be made using the Smith and Vanneste (2012) theory—which assumes that the D_j are positive—because γ_j(k) can be negative in certain propagation directions.

The Rhines modes of section 3b offer a basis of L² ⊕ ℂ that corresponds to Rossby waves over topography in the linear limit. These Rhines modes do not contain any free parameters. Indeed, if we set D₂ = 0 in the eigenvalue problem (80) and let D₁ = γ₁, we then obtain the Rhines modes. Note that since D₁ = γ₁ = γ₁(k) may be negative, the Smith and Vanneste (2012) modes do not apply. Instead, the case of negative D_j is examined in this article and in Yassin (2021).

However, the Rhines modes, as a basis of L² ⊕ ℂ, are not a basis of the whole vertical structure phase space L² ⊕ ℂ² since they exclude surface buoyancy anomalies at the upper boundary. To solve this problem, we can use the modes of the eigenvalue problem Eq. (80) with D₁ = γ₁ but leaving D₂ arbitrary as in Smith and Vanneste (2012). Although this basis now only has one free parameter, D₂, it still does not correspond to Rossby waves in the linear limit. We can even eliminate this free parameter by interpreting surface buoyancy gradients as topography—for example, by defining

g_{buoy} = {(\frac{f_{0}^{2}}{N^{2}} \frac{d ψ_{B}}{d z})}_{z = z_{2}},

(91)

where ψ_B corresponds to the background flow, and using g_buoy in place of g₂ in the generalized Rhines modes of section 3c. However, the waves resulting from topographic gradients generally differ from those resulting from vertically sheared mean flows (in particular, one must take into account advective continuum modes) and so this resolution is artificial.

Galerkin approximations with L² modes

Both the L² baroclinic modes and the L² ⊕ ℂ² modes have infinitely many degrees of freedom. In contrast, numerical simulations only contain a finite number of degrees of freedom. Consequently, it should be possible to use baroclinic modes to produce a Galerkin approximation to quasigeostrophic theory with nontrivial boundary dynamics. Such an approach has been proposed by Rocha et al. (2015).

Projecting ψ_k onto the baroclinic modes produces a series expansion,

ψ_{k}^{exp}

, that is differentiable in the interior but not at the boundaries. By differentiating the series in the interior we obtain Eq. (77) for q_k_n. If instead we integrate by parts twice and avoid differentiating

ψ_{k}^{exp}

, we obtain

q_{k n} = - (k^{2} + λ_{n}) ψ_{k n} - \frac{1}{H} \sum_{j = 1}^{2} r_{j k} F_{n} (z_{j}) .

(92)

The two expressions Eqs. (77) and (92) are only equivalent when r₁_k = r₂_k = 0. For nonzero r₁_k and r₂_k, the singular nature of the expansion means we have a choice between Eqs. (77) and (92).

By choosing Eq. (92) and avoiding the differentiation of $ψ_{k}^{exp}$ , Rocha et al. (2015) produced a least squares approximation to quasigeostrophic dynamics that conserves the surface potential enstrophy integrals in Eq. (26). This is a conservation property underlying their approximation’s success.

6. Conclusions

In this article, we have studied all possible noncontinuum collections of quasigeostrophic streamfunction normal modes that diagonalize the energy and potential enstrophy. There are four possible L² modes: the baroclinic modes, the antibaroclinic modes, the surface modes, and the antisurface modes. Additionally, we explored the properties of the family of L² ⊕ ℂ² bases introduced by Smith and Vanneste (2012) that contain two free parameters D₁ and D₂ and generalized the family to allow for D₁, D₂ < 0. This generalization is necessary for Rossby waves in the presence of bottom topography. If D_j = γ_j, where γ_j is given by Eq. (44) for j = 1, 2, the resulting modes are the vertical structure of Rossby waves in a quiescent ocean with prescribed boundary buoyancy gradients (i.e., topography). We have also examined the associated L² and L² ⊕ ℂ² vertical velocity modes.

For the streamfunction L² modes, only the baroclinic modes are capable of converging pointwise to any quasigeostrophic state on the interval [z₁, z₂], whereas for the vertical velocity L² modes, only the antibaroclinic modes are capable. However, in both cases, the resulting eigenfunction expansion is not differentiable at the boundaries, z = z₁, z₂. Consequently, while we can recover the volume potential vorticity density q_k, we cannot recover the surface potential vorticity densities r₁_k and r₂_k. Thus, we lose two degrees of freedom when projecting onto the baroclinic modes. In contrast, L² ⊕ ℂ² modes provide an equivalent representation of the function in question. Namely, the eigenfunction expansion is differentiable on the closed interval [z₁, z₂] so that we can recover q_k, r₁_k, and r₂_k from the series expansion.

We have also introduced a new set of modes, the Rhines modes, that form a basis of L² ⊕ ℂ and correspond to the vertical structures of Rossby waves over topography. A natural application of these normal modes is to the study of weakly nonlinear wave-interaction theories of geostrophic turbulence found in Fu and Flierl (1980) and Smith and Vallis (2001), extending their work to include bottom topography.

A collection of functions is said to be complete in some function space Φ if this collection spans the space Φ. Specifying the underlying function space Φ turns out to be crucial, as we see in section 2d.

Continuum modes appear once a sheared mean flow is present [e.g., Drazin et al. (1982), Balmforth and Morrison (1994, 1995), and Brink and Pedlosky (2019)].

The definition of L²[D] is more subtle than is presented here. Namely, elements of L²[D] are not functions but rather are equivalence classes of functions leading to the unintuitive properties seen in this section. See Yassin (2021) and citations within for more details.

Since all physical fields must be real, only a single degree of freedom is gained from ℂ. Furthermore, when complex notation is used (e.g., complex exponentials for the horizontal eigenfunctions e_k) it is only the real part of the fields that is physical.

To apply the theory of Yassin (2021), summarized in appendix A, let $\tilde{λ} = λ - k^{2}$ be the eigenvalue in place of λ; the resulting eigenvalue problem for $\tilde{λ}$ will then satisfy the positiveness conditions, Eqs. (A7) and (A8), of appendix A.

That γ₁ is not positive prevents us from applying the eigenvalue theory outlined in the appendix of Smith and Vanneste (2012).

A pure surface quasigeostrophic mode is the mode found after setting β = 0 with an upper boundary at z₂ = ∞.

Recall that the geostrophic streamfunction ψ is proportional to pressure (e.g., Vallis 2017, his section 5.4).

In particular, chapters 1 and 8 in Levitan and Sargsjan (1975) show that eigenfunction expansions have the same pointwise convergence and differentiability properties as the Fourier series with the analogous boundary conditions. The behavior of Fourier series is discussed in Brown and Churchill (1993).

To see that $ψ_{k}^{exp}$ is nondifferentiable at z = z₁, z₂, suppose that the series $ψ_{k}^{exp}$ is differentiable and that dψ_k(z_j)/dz ≠ 0 for j = 1, 2; but then $0 \neq d ψ_{k} (z_{j}) / d z = \sum_{n = 0}^{\infty} ψ_{k n} [d F_{n} (z_{j}) / d z] = 0$ , which is a contradiction.

Acknowledgments.

We offer sincere thanks to Stephen Garner, Robert Hallberg, Isaac Held, Sonya Legg, and Shafer Smith for comments and suggestions that greatly helped our presentation. We also thank Guillaume Lapeyre, William Young, one anonymous reviewer, and the editor (Joseph LaCasce) for their comments that helped us to further refine and focus the presentation, and to correct confusing statements. This report was prepared by Houssam Yassin under Award NA18OAR4320123 from the National Oceanic and Atmospheric Administration of the U.S. Department of Commerce. The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the views of the National Oceanic and Atmospheric Administration or the U.S. Department of Commerce.

Data availability statement.

The data that support the findings of this study are available within the article.

APPENDIX A

Sturm–Liouville Eigenvalue Problems with λ-Dependent Boundary Conditions

Consider the differential eigenvalue problem given by Eqs. (4) and (5). We assume that 1/p(z), q(z), and r(z) are real-valued integrable functions and that a_j, b_j, c_j, and d_j are real numbers for j = 1, 2. Moreover, we assume that p > 0, r > 0, p and r are twice continuously differentiable, q is continuous, and (a_j, b_j) ≠ (0, 0).

Define the two boundary parameters D_j for j = 1, 2 by

D_{j} = {(- 1)}^{j + 1} (a_{j} d_{j} - b_{j} c_{j}) .

(A1)

Then the natural inner product for the eigenvalue problem is given by

〈 F, G 〉 = \int_{z_{1}}^{z_{2}} F G d z + \sum_{j = 1}^{2} D_{j}^{- 1} (C_{j} F) (C_{j} G)

(A2)

where the boundary operator Χ_j is defined by

C_{j} F = c_{j} F (z_{j}) - d_{j} (p \frac{d F}{d z}) (z_{j}) .

(A3)

The eigenvalue problem takes place in the space L² ⊕ ℂ^N, where N is the number of nonzero D_j. Assume for the following that N = 2; the case in which N = 1 is similar. If

D_{j} > 0

(A4)

for j = 1, 2 then the inner product (A2) is positive definite—that is, all nonzero F satisfy 〈F, F〉 > 0. Therefore L² ⊕ ℂ², equipped with the inner product Eq. (A2), is a Hilbert space. In this Hilbert-space setting, the eigenfunctions

{F_{n}}_{n = 0}^{\infty}

form an orthonormal basis of L² ⊕ ℂ² and the eigenvalues are distinct and bounded below as in Eq. (45) (Evans 1970; Walter 1973; Fulton 1977). The appendix of Smith and Vanneste (2012) also proves this result in the case in which d₁ = d₂ = 0. The convergence properties of normal-mode expansions in this case are due to Fulton (1977).

However, as we observe in section 3, the D_j > 0 case is not sufficient for the Rossby wave problem with topography. In general, the space L² ⊕ ℂ² with the indefinite inner product Eq. (A2) is a Pontryagin space (see Iohvidov and Krein 1960; Bognár 1974). Pontryagin spaces are analogous to Hilbert spaces except that they have a finite-dimensional subspace of elements satisfying 〈F, F〉 < 0. If Π is a Pontryagin space with inner product 〈,〉, then Π admits a decomposition

Π = Π^{+} \oplus Π^{-},

(A5)

where Π⁺ is a Hilbert space under the inner product 〈,〉 and Π⁻ is a finite-dimensional Hilbert space under the inner product −〈,〉. If {G_n}_n₌₀ is an orthonormal basis for the Pontryagin space Π, then an element Ψ ∈ Π can be expressed

Ψ = \sum_{n = 0}^{\infty} \frac{〈 Ψ, G_{n} 〉}{〈 G_{n}, G_{n} 〉} .

(A6)

Even though

{G_{n}}_{n = 0}^{\infty}

is normalized, the presence of 〈G_n, G_n〉 = ±1 in the denominator of Eq. (A6) is essential since this term may be negative.

One can rewrite the eigenvalue problem in Eqs. (4)–(5) in the form LF = λF for some operator L (e.g., Langer and Schneider 1991). The operator L is a positive operator if for the λ-dependent boundary conditions we have

\frac{a_{i} c_{i}}{D_{i}} \leq 0, \frac{b_{i} d_{i}}{D_{i}} \leq 0, and {(- 1)}^{i} \frac{a_{i} d_{i}}{D_{i}} \geq 0

(A7)

or for the λ-independent boundary conditions we have

b_{i} = 0 or {(- 1)}^{i + 1} \frac{a_{i}}{b_{i}} \geq 0 if b_{i} \neq 0.

(A8)

Yassin (2021) has shown that, when L is positive, the eigenfunctions

{F_{n}}_{n = 0}^{\infty}

of the eigenvalue problem in Eqs. (4)–(5) form an orthonormal basis of L² ⊕ ℂ²; that the eigenvalues are real; and that the eigenvalues are ordered as in Eq. (45). Moreover, since L is positive, we have the relationship

λ 〈 F, F 〉 = 〈 L F, F 〉 \geq 0.

(A9)

Yassin (2021) also shows that the normal-mode-expansion results of Fulton (1977) extend to this case as well.

APPENDIX B

Polarization Relations and the Vertical Velocity Eigenvalue Problem

a. Polarization relations

The linear quasigeostrophic vorticity and buoyancy equations, computed about a resting background state, are

\frac{\partial ζ}{\partial t} + β \frac{\partial ψ}{\partial x} = f_{0} \frac{\partial w}{\partial z} and

(B1)

\frac{\partial b}{\partial t} = - N^{2} w

(B2)

in the interior z ∈ (z₁, z₂). The vorticity ζ and buoyancy b are given in terms of the geostrophic streamfunction by

ζ = \nabla^{2} ψ and

(B3)

b = f_{0} \frac{\partial ψ}{\partial z},

(B4)

The no-normal flow at the lower and upper boundaries implies

f_{0} w = u \cdot g_{j}

(B5)

for j = 1, 2. Substituting Eq. (B5) into the linear buoyancy Eq. (B2), yields the boundary conditions

\partial_{t} b + u \cdot \nabla (\frac{N^{2}}{f_{0}} g_{j}) = 0 for z = z_{j} .

(B6)

We now assume solutions of the form

ψ = \hat{ψ} (z) e_{k} (x) e^{- i ω t}

(B7)

and similarly for w. Substituting such solutions into Eqs. (B1) and (B2) and using

u = \hat{z} \times \nabla ψ

give

\frac{d \hat{ψ}}{d z} = - i \frac{N^{2}}{f_{0} ω} \hat{w} and

(B8)

\frac{d \hat{w}}{d z} = i \frac{ω}{f_{0}} (k^{2} + \frac{β k_{x}}{ω}) \hat{ψ} .

(B9)

for z ∈ (z₁, z₂). At the boundaries z = z₁ and z₂, we use Eqs. (B5) and (B6) to obtain

\hat{b} = - \frac{N^{2}}{f_{0} ω} \hat{u} \cdot \nabla g_{j} and

(B10)

\hat{w} = i \frac{1}{f_{0}} \hat{u} \cdot \nabla g_{j} .

(B11)

b. The vertical velocity eigenvalue problem

Taking the vertical derivative of Eq. (B9) and using Eq. (B8) yield

- \frac{d^{2} χ}{d z^{2}} = λ (\frac{N^{2}}{f_{0}^{2}}) χ,

(B12)

where

\hat{w} = w_{0} χ (z)

and χ is nondimensional. The boundary conditions at z = z_j are

- {(- 1)}^{j} γ_{j} k^{2} \frac{d χ}{d z} = λ [χ + {(- 1)}^{j} γ_{j} \frac{d χ}{d z}]

(B13)

as obtained by using Eqs. (B9) and (B8) in boundary conditions Eq. (55b). The orthonormality condition is

\pm δ_{m n} = \frac{1}{H} [\int_{z_{1}}^{z_{2}} χ_{m} χ_{n} (\frac{N^{2}}{f_{0}^{2}}) d z - \frac{1}{k^{2}} \sum_{j = 1}^{2} \frac{1}{γ_{j}} (C_{j} χ_{m}) (C_{j} χ_{n})],

(B14)

where

C_{j} χ = χ (z_{j}) + {(- 1)}^{j} γ_{j} \frac{d χ (z_{j})}{d z} .

(B15)

When only one boundary condition is λ dependent (e.g., γ₂ = 0), the eigenvalue problem in Eqs. (B12) and (B13) satisfies Eq. (A4) when γ₁ > 0 and Eqs. (A7) and (A8) when γ₁ < 0; thus, the reality of the eigenvalues and the completeness results follow. However, when both boundary conditions are λ dependent the problem no longer satisfies these conditions for all k. Instead, in this case, one exploits the relationship between the vertical velocity eigenvalue problem in Eqs. (B12) and (B13) and the streamfunction problem in Eqs. (55a) and (55b) given by Eqs. (B8) and (B9) to conclude that the two problems have identical eigenvalues (for ω ≠ 0), and then one uses the simplicity of the eigenvalues to conclude that no generalized eigenfunctions can arise.

c. The vertical velocity L² modes

Analogous to the streamfunction L² modes, we have the following sets of vertical velocity L² modes: For baroclinic modes, we have vanishing vertical velocity at both boundaries,

χ (z_{1}) = 0 and χ (z_{2}) = 0.

(B16)

For antibaroclinic modes, we have vanishing pressure at both boundaries,

\frac{d χ (z_{1})}{d z} = 0 and \frac{d χ (z_{2})}{d z} = 0.

(B17)

For surface modes, we have

\frac{d χ (z_{1})}{d z} = 0 and χ (z_{2}) = 0.

(B18)

For antisurface modes, we have

χ (z_{1}) = 0 and \frac{d χ (z_{2})}{d z} = 0.

(B19)

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Journal of Physical Oceanography

Abstract
1. Introduction
2. Mathematics of the quasigeostrophic phase space
3. Rossby waves in a quiescent ocean
4. Eigenfunction expansions
5. Discussion
- Galerkin approximations with L2 modes
6. Conclusions
Acknowledgments.
Data availability statement.
APPENDIX A
- Sturm–Liouville Eigenvalue Problems with λ-Dependent Boundary Conditions
APPENDIX B
- Polarization Relations and the Vertical Velocity Eigenvalue Problem
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On the Discrete Normal Modes of Quasigeostrophic Theory

Abstract

Abstract

1. Introduction

a. Background

b. Normal modes and eigenfunctions

c. Contents of this article

2. Mathematics of the quasigeostrophic phase space

a. The potential vorticity

b. Defining the quasigeostrophic phase space

c. The phase space in terms of the streamfunction

d. The vertical structure phase space

e. Representing the energy and potential enstrophy

3. Rossby waves in a quiescent ocean

a. Traditional Rossby wave problem

1) Traditional baroclinic modes

2) Stationary step modes

3) The general solution

b. The Rhines problem

1) The eigenvalue problem

2) Characterizing the eigensolutions

3) The general time-dependent solution

c. The generalized Rhines problem

d. The vertical velocity eigenvalue problem

Quasigeostrophic boundary dynamics

4. Eigenfunction expansions

a. The four possible L2 modes

b. Expansions with L2 modes

1) Pointwise equality on [z1, z2]

2) Differentiability of the series expansion

c. Quasigeostrophic L2 ⊕ ℂ2 modes

d. Expansion with L2 ⊕ ℂ2 modes

5. Discussion

Galerkin approximations with L2 modes

6. Conclusions

Acknowledgments.

Data availability statement.

APPENDIX A

Sturm–Liouville Eigenvalue Problems with λ-Dependent Boundary Conditions

APPENDIX B

Polarization Relations and the Vertical Velocity Eigenvalue Problem

a. Polarization relations

b. The vertical velocity eigenvalue problem

c. The vertical velocity L2 modes

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a. The four possible L² modes

b. Expansions with L² modes

1) Pointwise equality on [z₁, z₂]

c. Quasigeostrophic L² ⊕ ℂ² modes

d. Expansion with L² ⊕ ℂ² modes

Galerkin approximations with L² modes

c. The vertical velocity L² modes