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  • Yassin, H., 2021: Normal modes with boundary dynamics in geophysical fluids. J. Math. Phys., 62, 093102, https://doi.org/10.1063/5.0048273.

  • View in gallery

    Polar plots of the absolute value of the nondimensional angular frequency |ωn|/(βLd) 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 kLd = 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 Ld = N0H/f0 = 100 km. Numerical solutions to all eigenvalue problems in this paper are obtained using Dedalus (Burns et al. 2020).

  • View in gallery

    Polar plots of the absolute value of the nondimensional angular frequency |ωn|/(βLd) of the first five modes from section 3b as a function of the wave propagation direction k/|k| for a horizontal wavenumber given by kLd = 7 in constant stratification. The dashed line corresponds to ω0; this mode becomes boundary trapped at large wavenumbers k = |k|. The remaining modes, ωn 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 ωn and so have a westward phase speed. Gray regions are angles where γ1 < 0. Here, ω0 is positive while the remaining angular frequencies ωn 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.

  • View in gallery

    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 kLd = 1. The phase speed approaches zero in the limit γ1 → 0. (b) Here, γ1/H ≈ 10 for the three vertical structures Gn shown. Consequently, the bottom trapped wave has λ ≈ −k2 and the phase speeds are large. The vertical structure G for three values of kLd 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 = k1. In both (a) and (b), the wave propagation direction θ = 260°. All other parameters are identical to Fig. 2.

  • View in gallery

    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, ωn 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 ωn 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 ωn 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.

  • View in gallery

    This figure illustrates the dependence of the vertical structure Gn 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 kLd = (left) 0.5 and (right) 7 (where k = |k|) [e.g., (b), (d), and (f) are the vertical structure of waves with kLd = 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 kLd = 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.

  • View in gallery

    The first six vertical velocity normal modes χn (thin gray lines) and streamfunction normal modes Gn (black lines) (see section 3d). The propagation direction is θ = 75° with a wavenumber of kLd = 2. The remaining parameters are as in Fig. 2. Note that χn and Gn 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 Ld.

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On the Discrete Normal Modes of Quasigeostrophic Theory

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  • 1 aProgram in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey
  • | 2 bNOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey
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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.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Houssam Yassin, hyassin@princeton.edu

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.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Houssam Yassin, hyassin@princeton.edu

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 complete1 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
Φa(x,y,z)eiωat,
where Φa describes the spatial structure of the mode and ωa is its angular frequency. The function Φa 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, Φa, is often called a normal mode (e.g., the term “Fourier mode” is often used for eikx, where k is a wavenumber). In linear theory, this misnomer is often benign because each Φa corresponds to a frequency ωa. 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)=acaΦa(x,y,z),
where the ca are the Fourier coefficients, and the time evolution is then given by
acaΦa(x,y,z)eiωat.

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 Φa 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 Φa, which is only an eigenfunction. Otherwise, we will use the terms “normal mode” and “eigenfunction” interchangeably to refer to the spatial structure Φa, 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 Fn(z). These vertical normal modes are obtained by solving a differential eigenvalue problem of the form
ddz(pdFdz)+qF=λrF
in the interval (z1, z2) with boundary conditions
[ajFbj(pdFdz)(zj)]=λ[cjF(zj)dj(pdFdz)(zj)]
for j = 1, 2, where 1/p(z), q(z), and r(z) are real-valued functions and aj, bj, cj, and dj 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 Fn (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., noncontinuum2) 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=D0×(z1,z2).
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+2ψ+z(f02N2ψz).
Here, the latitude-dependent Coriolis parameter is
f=f0+βy,
N(z) is the prescribed background buoyancy frequency, ∇2 is the horizontal Laplacian operator, and
u=z^×ψ
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
Rj=(1)j+1[gj+(f02N2ψz)|z=zj]
where gj is an imposed surface potential vorticity density at the lower or upper boundary and j = 1, 2. The density gj corresponds to a prescribed buoyancy
bj=N2f0gj
at the jth boundary [see Eq. (B6)]. Alternatively, gj may be thought of as an infinitesimal topography through
gj=f0hj
where hj represents infinitesimal topography at the jth boundary. Whereas Q has dimensions of inverse time, Rj 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 Rj = Rj(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 functions3 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, R2L2[D0].

Define the space P by
P=L2[D]L2[D0]L2[D0],
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 Rj = Rj(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
Qf=2ψint+z(f0N2ψintz)forz(z1,z2),
R1g1=f02N2ψlowzforz=z1
R2+g2=f02N2ψuppzforz=z2,
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 = Qf and rj = Rj − (−1)j+1gj, which are the portions of the potential vorticity providing a source for a streamfunction. Explicitly, the relative potential vorticity densities are
q=2ψ+z(f02N2ψz)forz(z1,z2),
r1=f02N2ψzforz=z1,and
r2=f02N2ψzforz=z2.

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 ek of the horizontal Laplacian. For the rectangular horizontal domain D0, the eigenfunction ek(x) satisfies
2ek=k2ek.
where x = (x, y) is the horizontal position vector, k = (kx, ky) is the horizontal wavevector, and k = |k| is the horizontal wavenumber. For example, in a horizontally periodic domain the eigenfunctions ek(x) are proportional to complex exponentials eikx.
Projecting the relative potential vorticity density distribution, (q, r1, r2), onto the horizontal eigenfunctions ek yields
q(x,z,t)=kqk(z,t)ek(x),for z(z1,z2),and
rj(x,t)=krjk(t)ek(x)for j=1,2.
Thus the Fourier coefficients of (q, r1, r2) are (qk, r1k, r2k) where qk is a function of z, and r1k and r2k are independent of z. Hence, qk is an element of L2[(z1, z2)] whereas r1k and r2k are elements of the space of complex numbers4 ℂ.
We conclude that the vertical structure of the potential vorticity, given by (qk, r1k, r2k), is an element of
P^=L2[(z1,z2)],
so that the vertical structures of the potential vorticity distribution are determined by a function qk in L2[(z1, z2)] and two z-independent elements r1k and r2k of ℂ. Similarly, the streamfunction can be represented as
ψ(x,z,t)=kψk(z,t)ek(x),
where ψk and (qk, r1k, r2k) are related by
qk=k2ψk+z(f02N2ψkz)and
rjk=(1)j+1(f02N2ψkz)|z=zj.

As before, knowledge of the vertical structure of the streamfunction ψk(z) is equivalent to knowing the vertical structure of the potential vorticity distribution (qk, r1k, r2k). In the resulting differential eigenvalue problem for the vertical normal modes, the nonzero rjk 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 P^ given by Eq. (18) (Yassin 2021). Thus P^ is also the space of all possible streamfunction vertical structures.

That ψk belongs to 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 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 P^ first appeared in Smith and Vanneste (2012). More generally, P^ appears in the presence of nontrivial boundary dynamics (Yassin 2021).

We call 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 P^ is then written as the direct sum
P^=L22.

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 ek(x). The energy per unit mass in the volume D is given by
E=1VD[|ψ|2+f02N2|ψz|2]dAdz=kEk,
where the horizontal energy mode is given by the vertical integral
Ek=1Hz1z2[k2|ψk|2+f02N2|ψkz|2]dz,
with V = AH being the domain volume and H = z2z1 being the domain depth.
Similarly, for the relative volume potential enstrophy density Z, we have
Z=1VD|q|2dAdz=kZk
where
Zk=1Hz1z2|qk|2dz.
Last, analogous to Z, we have the relative surface potential enstrophy densities Yj on the area D0:
Yj=1AD0|rj|2dA=kYjk
where
Yjk=|rjk|2.

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
qt+βυ=0for z(z1,z2)and
rjt+u·[(1)j+1gj]=0for z=zj.
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)=ψ^(z)ek(x)eiωt
into the linear problem Eq. (28) renders
(iω)[k2ψ^+ddz(f02N2dψ^dz)]+ikxβψ^=0
for z ∈ (z1, z2), and
(iω)(f02N2dψ^dz)+iz^·(k×gj)ψ^=0
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[k2F+ddz(f02N2dFdz)]βkxF=0and
ω(f02N2dFdz)|z=zj=0,
where ψ^(z)=ψ^0F(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:
ddz(f02N2dFdz)=λFfor z(z1,z2)and
f02N2dFdz=0for z=z1,z2,
where the eigenvalue λ is given by
λ=k2βkxω.
See Fig. 1 for an illustration of the dependence of |ω| on the wavevector k.
Fig. 1.
Fig. 1.

Polar plots of the absolute value of the nondimensional angular frequency |ωn|/(βLd) 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 kLd = 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 Ld = N0H/f0 = 100 km. Numerical solutions to all eigenvalue problems in this paper are obtained using Dedalus (Burns et al. 2020).

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

Fig. 2.
Fig. 2.

Polar plots of the absolute value of the nondimensional angular frequency |ωn|/(βLd) of the first five modes from section 3b as a function of the wave propagation direction k/|k| for a horizontal wavenumber given by kLd = 7 in constant stratification. The dashed line corresponds to ω0; this mode becomes boundary trapped at large wavenumbers k = |k|. The remaining modes, ωn 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 ωn and so have a westward phase speed. Gray regions are angles where γ1 < 0. Here, ω0 is positive while the remaining angular frequencies ωn 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.

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 λn satisfying
0=λ0<λ1<.
The nth mode, Fn, 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]=1Hz1z2FGdz,
with orthonormality, meaning that
δmn=[Fm,Fn],
where δmn is the Kronecker delta. A powerful and commonly used result of Sturm–Liouville theory is that the set {Fn}n=0 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
βkxF=0for z(z1,z2)and
0=0for z=z1,z2.
Consequently, if kx ≠ 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
Fjstep(z)={1for z=zj0otherwise.
The two step-mode solutions in Eq. (39) are independent of the traditional baroclinic modes Fn(z). An expansion of the step mode Fjstep in terms of the baroclinic modes will fail and produce a series that is identically zero.

The two stationary step modes, F1step and F2step, 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 N2gj/f0 become small.

3) The general solution

For a wavevector k with kx ≠ 0, the vertical structure of the streamfunction must be of the form
Ψ(z)+j=12ΨjFjstep(z)=ψk(z,t=0),
where Ψ(z) is a twice differentiable function satisfying dΨ(zj)/dz = 0 for j = 1, 2 and Ψ1 and Ψ2 are arbitrary constants. We can represent Ψ according to the expansion
Ψ=n=0[Ψ,Fn]Fn,
and so the time evolution is
ψk(z,t)=n=0[Ψ,Fn]Fneiωnt+j=12ΨjFjstep.
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 Fn. 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 ψ^(z)=ψ^0G(z), where G is a nondimensional function. We then manipulate the eigenvalue problem in Eqs. (30)(31) to obtain (assuming ω ≠ 0)
ddz(f02N2dGdz)=λGfor z(z1,z2)
k2Gγ11(f02N2dGdz)=λGfor z=z1,and
dGdz=0for z=z2,
where the length scale γj is given by
γj=(1)j+1z^·(k×gj)z^·(k×f)=(1)j+1(αjkβkx)sin(Δθj),
where αj = |∇gj| and Δθj is the angle between the wavevector k and ∇gj measured counterclockwise from k. The parameter γj depends only on the direction of the wavevector k and not its magnitude k = |k|. If γj = 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 λn satisfying
λ0<λ1<λ2<.
The inner product 〈,〉 induced by the eigenvalue problem Eq. (43) is
F,G=1H[z1z2FGdz+γ1F(z1)G(z1)],
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 Gn then takes the form
±δmn=Gm,Gn,
where at most one mode, Gn, satisfies 〈Gn, Gn〉 = −1. The eigenfunctions {Gn}n=0 form an orthonormal basis of L2 ⊕ ℂ under the inner product Eq. (46).
Appendix A provides the following inequality,
(k2+λn)Gn,Gn>0,
that, using the dispersion relation in Eq. (34), implies that modes Gn with 〈Gn, Gn〉 > 0 correspond to waves with a westward phase speed whereas modes Gn with 〈Gn, Gn〉 < 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.

  1. γ1 > 0: All eigenvalues satisfy λn > −k2, all modes satisfy 〈Gn, Gn〉 > 0, and all waves propagate westward. The nth mode Gn has n internal zeros (Binding et al. 1994). See the regions in white in Fig. 2.

  2. γ1 < 0: There is one mode, G0, with a negative square, 〈Gn, Gn〉 < 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 λn < −k2. The remaining modes, Gn for n > 1, have positive squares, 〈Gn, Gn〉 > 0, corresponding to westward-propagating waves and have eigenvalues λn satisfying λn > −k2. Both G0 and G1 have no internal zeros, whereas the remaining modes, Gn, have n − 1 internal zeros for n > 1 (Binding et al. 1994). See the stippled regions in Fig. 2.

To elucidate the meaning of λn < −k2, note that a pure surface quasigeostrophic mode7 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
(f02N2dGndz)|z=z1=0.
Fig. 3.
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 kLd = 1. The phase speed approaches zero in the limit γ1 → 0. (b) Here, γ1/H ≈ 10 for the three vertical structures Gn shown. Consequently, the bottom trapped wave has λ ≈ −k2 and the phase speeds are large. The vertical structure G for three values of kLd 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 = k1. In both (a) and (b), the wave propagation direction θ = 260°. All other parameters are identical to Fig. 2.

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(z1)=0
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)=ψk(z,t=0)
where Ψ is a twice continuously differentiable function satisfying dΨ(z2)/dz = 0. For such functions we can write (see appendix A)
Ψ=n=0Ψ,GnGn,GnGn.
so that the time evolution is
ψk(z,t)=n=0Ψ,GnGn,GnGn(z)eiωnt.
Again, it is the above expression, which is valid only in linear theory with a quiescent background state, that gives the generalized Rhines modes Gn 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 {Fn}n=0 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, F1step(z), in terms of the L2 ⊕ ℂ1 modes, {Gn}n=0, with the expansion given by
F1step(z)=γ1Hn=0Gn(z1)Gn,GnGn(z).

c. The generalized Rhines problem

The general problem with topography at both the upper and lower boundaries is
ddz(f02N2dGdz)=λGfor z(z1,z2)
k2G+(1)jγj1(f02N2dGdz)=λGfor z=zj,
for j = 1, 2, where the length scale γj 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=1H[z1z2FGdz+j=12γjF(zj)G(zj)],
which reduces to Eq. (46) when γ2 = 0. Under this inner product, the eigenfunctions {Gn}n=0 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.
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, ωn 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 ωn 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 ωn 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.

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

Fig. 5.
Fig. 5.

This figure illustrates the dependence of the vertical structure Gn 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 kLd = (left) 0.5 and (right) 7 (where k = |k|) [e.g., (b), (d), and (f) are the vertical structure of waves with kLd = 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 kLd = 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.

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

  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.

  2. γ1γ2 < 0: This corresponds to case ii in section 3b. See the stippled regions in Fig. 4 and Figs. 5c and 5d.

  3. γ1 < 0 and γ2 < 0: There are two modes G0 and G1 with negative squares, 〈Gn, Gn〉 < 0, that propagate eastward and have eigenvalues Gn satisfying Gn < −k2 for n = 1, 2. The remaining modes Gn for n > 1 have positive squares, 〈Gn, Gn〉 > 0, propagate westward, and have eigenvalues, λn, satisfying λn > −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 Gn 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 w^(z)=w^0χ(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
d2χdz2=λ(N2f02)χ
with vanishing vertical velocity boundary conditions
χ(zj)=0
(see appendix B for details). The resulting modes {χn}n=0 form an orthonormal basis of L2 with orthonormality given by
δmn=1Hz1z2χmχn(N2f02)dz.
One can obtain the eigenfunctions χn by solving the eigenvalue problem in Eqs. (57)(58) or by differentiating the streamfunction modes Fn 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)jγjk2dχdz|zj=λ[χ|zj+(1)jγjdχdz|zj]
(see appendix B). The resulting modes {χn}n=0 (Fig. 6) satisfy a peculiar orthogonality relation given by Eq. (B14).
Fig. 6.
Fig. 6.

The first six vertical velocity normal modes χn (thin gray lines) and streamfunction normal modes Gn (black lines) (see section 3d). The propagation direction is θ = 75° with a wavenumber of kLd = 2. The remaining parameters are as in Fig. 2. Note that χn and Gn 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 Ld.

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 {Fn}n=0 be a collection of continuous functions that form a basis of L2 and assume ψk(z, t) is twice continuously differentiable in z. Define the eigenfunction expansion ψkexp of ψ by
ψkexp(z,t)=n=0ψkn(t)Fn(z),
where
ψkn=[ψk,Fn].
Because {Fn}n=0 is a basis of L2, the eigenfunction expansion ψkexp satisfies (e.g., Brown and Churchill 1993)
z1z2|ψk(z)ψkexp(z)|2dz=0.
Significantly, the vanishing of the integral in Eq. (63) does not imply ψk(z)=ψkexp(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
ddz(f02N2dFdz)=λFz(z1,z2),
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),

dF(z1)dz=0anddF(z2)dz=0,
  • antibaroclinic modes: these have vanishing pressure8 at both boundaries (Dirichlet),

F(z1)=0andF(z2)=0,
  • surface modes: (mixed Neumann/Dirichlet)

F(z1)=0anddF(z2)dz=0,and
  • antisurface modes: (mixed Neumann/Dirichlet)

dF(z1)dz=0andF(z2)=0.
All four sets of modes are missing two modes. Each boundary condition of the form
dF(zj)dz=0
implies a missing step mode, and a boundary condition of the form
F(zj)=0
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 ψk is twice continuously differentiable in z, we obtain pointwise equality in the interior
ψk(z)=ψkexp(z)for z(z1,zz).
The behavior at the boundaries depends on the boundary conditions that the modes Fn satisfy. If the Fn satisfy the vanishing pressure boundary condition at the jth boundary
Fn(zj)=0,
then
ψkexp(zj)=0
regardless of the values of ψk(zj). It follows that ψkexp will be continuous over (z1, z2) and will generally have a jump discontinuity at the boundaries [unless ψk(zj) = 0 for j = 1, 2]. In contrast, if the Fn satisfy a zero vertical velocity boundary condition at the jth boundary
dFn(zj)dz=0,
then
ψk(zj)=ψkexp(zj).
Consequently, of the four sets of L2 modes, only with the baroclinic modes do we obtain the pointwise equality ψk(z)=ψkexp(z) on the closed interval [z1, z2].

However, even though ψkexp converges pointwise to ψk when the baroclinic modes are used, we are unable to represent the corresponding velocity wk 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 qk is associated with the L2 degrees of freedom while the surface potential vorticities, r1k and r2k, are associated with the ℂ2 degrees of freedom.

The series expansion ψkexp of ψk in terms of the baroclinic modes is differentiable in the interior (z1, z2). Consequently, we can differentiate the ψkexp series for z ∈ (z1, z2) to recover qk; that is,
qk=n=0qknFn
where
qkn=(k2+λn)ψkn.
However, ψkexp is not differentiable at the boundaries z = z1, z2, so we are unable to recover the surface potential vorticities r1k and r2k. Two degrees of freedom are lost by projecting onto the baroclinic modes.10
The energy at wavevector k is indeed partitioned between the modes
Ek=n=0(k2+λn)|ψkn|2,
and similarly for the potential enstrophy
Zk=n=0(k2+λn)2|ψkn|2.
However, because we have lost r1k and r2k in the projection process, the surface potential enstrophies Y1k and Y2k, defined in Eq. (27), are not partitioned.

c. Quasigeostrophic L2 ⊕ ℂ2 modes

Consider the eigenvalue problem
ddz(f02N2dGdz)=λGfor z(z1,z2)and
k2G+(1)jDj1(f02N2dGdz)=λGfor z=zj,
where D1 and D2 are nonzero real constants. This eigenvalue problem differs from the generalized Rhines eigenvalue problem in Eq. (55) in that Dj are generally not equal to the γj defined in Eq. (44). The inner product 〈,〉 induced by the eigenvalue problem in Eq. (80) is given by Eq. (56) with the γj replaced by the Dj.

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 Dj = γj and γj 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 {Gn}n=0 form a basis for the vertical structure phase space L2 ⊕ ℂ2. Thus, the projection
ψkexp(z)=n=0ψknGn(z),
where
ψkn=ψk,GnGn,Gn
is an equivalent representation of ψk. Not only do we have pointwise equality
ψk(z)=ψkexp(z)for z[z1,z2],
but the series ψkexp is also differentiable on the closed interval [z1, z2] [the case of Dj > 0 is due to Fulton (1977) whereas the case of Dj < 0 is due to Yassin (2021)]. Thus, given ψkexp, we can differentiate to obtain both qk and rjk and thereby recover all quasigeostrophic degrees of freedom. Indeed, we have
qk(z,t)=n=0qkn(t)Gn(z)and
rjk(t)=n=0rjkn(t)Gn(zj)
where
qkn=(k2+λn)Ψ,GnGn,Gnand
rjkn=Djqkn
for j = 1, 2.
In addition, the energy Ek, volume potential enstrophy Zk, and surface potential enstrophies Y1k and Y2k are partitioned (diagonalized) between the modes
Ek=n=0Gn,Gn(k2+λn)|ψkn|2and
Zk+1Hj=121DjYjk=n=0Gn,Gn(k2+λn)2|ψkn|2.

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 Dj = γj(k) for each wavevector k], using the relationship between qkn and rjkn given by Eq. (87), the orthonormality condition Eq. (56), and assuming γj(k) ≠ 0, we obtain
dqkndt+iβkxψkn=0.
This expansion diagonalizes the linear terms (i.e., β and ∇gj) in the time-evolution Eqs. (28a) and (28b) through the choice Dj = γj(k) for each vector k. However, this choice cannot be made using the Smith and Vanneste (2012) theory—which assumes that the Dj are positive—because γj(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 Dj 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
gbuoy=(f02N2dψBdz)z=z2,
where ψB corresponds to the background flow, and using gbuoy 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 ψk onto the baroclinic modes produces a series expansion, ψkexp, that is differentiable in the interior but not at the boundaries. By differentiating the series in the interior we obtain Eq. (77) for qkn. If instead we integrate by parts twice and avoid differentiating ψkexp, we obtain
qkn=(k2+λn)ψkn1Hj=12rjkFn(zj).

The two expressions Eqs. (77) and (92) are only equivalent when r1k = r2k = 0. For nonzero r1k and r2k, 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 ψkexp, 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 Dj = γj, where γj 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 qk, we cannot recover the surface potential vorticity densities r1k and r2k. 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 qk, r1k, and r2k 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.

1

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.

2

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)].

3

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.

4

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 ek) it is only the real part of the fields that is physical.

5

To apply the theory of Yassin (2021), summarized in appendix A, let λ˜=λk2 be the eigenvalue in place of λ; the resulting eigenvalue problem for λ˜ will then satisfy the positiveness conditions, Eqs. (A7) and (A8), of appendix A.

6

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

7

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

8

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

9

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).

10

To see that ψkexp is nondifferentiable at z = z1, z2, suppose that the series ψkexp is differentiable and that k(zj)/dz ≠ 0 for j = 1, 2; but then 0dψk(zj)/dz=n=0ψkn[dFn(zj)/dz]=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 aj, bj, cj, and dj 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 (aj, bj) ≠ (0, 0).

Define the two boundary parameters Dj for j = 1, 2 by
Dj=(1)j+1(ajdjbjcj).
Then the natural inner product for the eigenvalue problem is given by
F,G=z1z2FGdz+j=12Dj1(CjF)(CjG)
where the boundary operator Χj is defined by
CjF=cjF(zj)dj(pdFdz)(zj).
The eigenvalue problem takes place in the space L2 ⊕ ℂN, where N is the number of nonzero Dj. Assume for the following that N = 2; the case in which N = 1 is similar. If
Dj>0
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 {Fn}n=0 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 Dj > 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+Π,
where Π+ is a Hilbert space under the inner product 〈,〉 and Π is a finite-dimensional Hilbert space under the inner product −〈,〉. If {Gn}n=0 is an orthonormal basis for the Pontryagin space Π, then an element Ψ ∈ Π can be expressed
Ψ=n=0Ψ,GnGn,Gn.
Even though {Gn}n=0 is normalized, the presence of 〈Gn, Gn〉 = ±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
aiciDi0,bidiDi0,and(1)iaidiDi0
or for the λ-independent boundary conditions we have
bi=0or(1)i+1aibi0 ifbi0.
Yassin (2021) has shown that, when L is positive, the eigenfunctions {Fn}n=0 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
λF,F=LF,F0.
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
ζt+βψx=f0wzand
bt=N2w
in the interior z ∈ (z1, z2). The vorticity ζ and buoyancy b are given in terms of the geostrophic streamfunction by
ζ=2ψand
b=f0ψz,
The no-normal flow at the lower and upper boundaries implies
f0w=u·gj
for j = 1, 2. Substituting Eq. (B5) into the linear buoyancy Eq. (B2), yields the boundary conditions
tb+u·(N2f0gj)=0for z=zj.
We now assume solutions of the form
ψ=ψ^(z)ek(x)eiωt
and similarly for w. Substituting such solutions into Eqs. (B1) and (B2) and using u=z^×ψ give
dψ^dz=iN2f0ωw^and
dw^dz=iωf0(k2+βkxω)ψ^.
for z ∈ (z1, z2). At the boundaries z = z1 and z2, we use Eqs. (B5) and (B6) to obtain
b^=N2f0ωu^·gjand
w^=i1f0u^·gj.

b. The vertical velocity eigenvalue problem

Taking the vertical derivative of Eq. (B9) and using Eq. (B8) yield
d2χdz2=λ(N2f02)χ,
where w^=w0χ(z) and χ is nondimensional. The boundary conditions at z = zj are
(1)jγjk2dχdz=λ[χ+(1)jγjdχdz]
as obtained by using Eqs. (B9) and (B8) in boundary conditions Eq. (55b). The orthonormality condition is
±δmn=1H[z1z2χmχn(N2f02)dz1k2j=121γj(Cjχm)(Cjχn)],
where
Cjχ=χ(zj)+(1)jγjdχ(zj)dz.

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(z1)=0andχ(z2)=0.
For antibaroclinic modes, we have vanishing pressure at both boundaries,
dχ(z1)dz=0anddχ(z2)dz=0.
For surface modes, we have
dχ(z1)dz=0andχ(z2)=0.
For antisurface modes, we have
χ(z1)=0anddχ(z2)dz=0.

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