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 boundarytrapped exponential surface quasigeostrophic solution (see Held et al. 1995) and suggests that the surface signal primarily reflects, not thermocline motion, but boundarytrapped 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
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.
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 surfaceaware 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 eigenvaluedependent 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 stepmode solutions that have not been noted before. The stationary step modes are the limits of boundarytrapped 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
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, R_{1}, and R_{2}. 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 R_{1} and R_{2} are only defined on the twodimensional lower and upper boundary surfaces D_{0} 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 L^{2}[D] be the space of squareintegrable functions^{3} in the fluid volume D and let L^{2}[D_{0}] be the space of squareintegrable functions on the boundary area D_{0}. Elements of L^{2}[D] are functions of three spatial coordinates, whereas elements of L^{2}[D_{0}] are functions of two spatial coordinates. Hence, Q ∈ L^{2}[D] and R_{1}, R_{2} ∈ L^{2}[D_{0}].
c. The phase space in terms of the streamfunction
d. The vertical structure phase space
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
That ψ_{k} belongs to
e. Representing the energy and potential enstrophy
3. Rossby waves in a quiescent ocean
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 ∇g_{1} = ∇g_{2} = 0 in the eigenvalue problem of Eqs. (30)–(31) gives
1) Traditional baroclinic modes
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 ω_{0}; 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 γ_{1} > 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 γ_{1} < 0. Here, ω_{0} is positive while the remaining angular frequencies ω_{n} for n > 0 are negative. Consequently, in the gray regions, ω_{0} 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 ∇g_{1}, points toward 55^{°} and corresponds to a bottom slope of ∇h_{1} = 1.5 × 10^{−5}, leading to γ_{1}/H = 0.15. The remaining parameters are as in Fig. 1.
Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPOD210199.1
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 ω_{0}; 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 γ_{1} > 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 γ_{1} < 0. Here, ω_{0} is positive while the remaining angular frequencies ω_{n} for n > 0 are negative. Consequently, in the gray regions, ω_{0} 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 ∇g_{1}, points toward 55^{°} and corresponds to a bottom slope of ∇h_{1} = 1.5 × 10^{−5}, leading to γ_{1}/H = 0.15. The remaining parameters are as in Fig. 1.
Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPOD210199.1
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 ω_{0}; 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 γ_{1} > 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 γ_{1} < 0. Here, ω_{0} is positive while the remaining angular frequencies ω_{n} for n > 0 are negative. Consequently, in the gray regions, ω_{0} 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 ∇g_{1}, points toward 55^{°} and corresponds to a bottom slope of ∇h_{1} = 1.5 × 10^{−5}, leading to γ_{1}/H = 0.15. The remaining parameters are as in Fig. 1.
Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPOD210199.1
2) Stationary step modes
The two stationary step modes,
3) The general solution
b. The Rhines problem
We now examine the case with a sloping lower boundary, ∇g_{1} ≠ 0, and an isentropic upper boundary, ∇g_{1} = 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 λdependent 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
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 L^{2} ⊕ ℂ.
2) Characterizing the eigensolutions
The following is obtained by applying the theory summarized in appendix A to the eigenvalue problem Eq. (43).^{5}
We distinguish the following cases depending on the sign of γ_{1}. In the following, we assume k ≠ 0.

γ_{1} > 0: All eigenvalues satisfy λ_{n} > −k^{2}, 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.

γ_{1} < 0: There is one mode, G_{0}, with a negative square, 〈G_{n}, G_{n}〉 < 0, corresponding to an eastwardpropagating wave. The eastwardpropagating wave nevertheless travels pseudowestward (to the left of the upslope direction for f_{0} > 0). The associated eigenvalue λ_{0} satisfies λ_{n} < −k^{2}. The remaining modes, G_{n} for n > 1, have positive squares, 〈G_{n}, G_{n}〉 > 0, corresponding to westwardpropagating waves and have eigenvalues λ_{n} satisfying λ_{n} > −k^{2}. Both G_{0} and G_{1} 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} < −k^{2}, note that a pure surface quasigeostrophic mode^{7} has λ = −k^{2}. Thus, λ_{0} < −k^{2} means that the bottomtrapped mode decays away from the boundary more rapidly than a pure surface quasigeostrophic wave. Indeed, the limit of λ_{0} → −∞ yields the bottom step mode (39) of section 3a(2).
The two limits of the boundarytrapped surface quasigeostrophic waves, as discussed in section 3c. (a) Convergence to the step mode given in Eq. (39) with j = 1 as γ_{1} → 0^{−} for three values of γ_{1} at a wavenumber k = k given by kL_{d} = 1. The phase speed approaches zero in the limit γ_{1} → 0^{−}. (b) Here, γ_{1}/H ≈ 10 for the three vertical structures G_{n} shown. Consequently, the bottom trapped wave has λ ≈ −k^{2} 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 boundarytrapped exponential mode with an efolding 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.
Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPOD210199.1
The two limits of the boundarytrapped surface quasigeostrophic waves, as discussed in section 3c. (a) Convergence to the step mode given in Eq. (39) with j = 1 as γ_{1} → 0^{−} for three values of γ_{1} at a wavenumber k = k given by kL_{d} = 1. The phase speed approaches zero in the limit γ_{1} → 0^{−}. (b) Here, γ_{1}/H ≈ 10 for the three vertical structures G_{n} shown. Consequently, the bottom trapped wave has λ ≈ −k^{2} 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 boundarytrapped exponential mode with an efolding 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.
Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPOD210199.1
The two limits of the boundarytrapped surface quasigeostrophic waves, as discussed in section 3c. (a) Convergence to the step mode given in Eq. (39) with j = 1 as γ_{1} → 0^{−} for three values of γ_{1} at a wavenumber k = k given by kL_{d} = 1. The phase speed approaches zero in the limit γ_{1} → 0^{−}. (b) Here, γ_{1}/H ≈ 10 for the three vertical structures G_{n} shown. Consequently, the bottom trapped wave has λ ≈ −k^{2} 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 boundarytrapped exponential mode with an efolding 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.
Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPOD210199.1
3) The general timedependent solution
c. The generalized Rhines problem
There are now three cases depending on the signs of γ_{1} and γ_{2} and as depicted in Figs. 4 and 5. In the following, we assume k ≠ 0.
As in Fig. 2, but now with an upper slope ∇h_{2} = 10^{−5} in the direction 200^{°} in addition to the bottom slope in Fig. 2. The upper slope corresponds to γ_{2}/H = 0.1. The dotted line corresponds to ω_{0} and the dashed line corresponds to ω_{1}, 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 γ_{1} > 0 and γ_{2} > 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 γ_{1} < 0 and γ_{2} < 0. The two gravest angular frequencies ω_{0} and ω_{1} are both positive while the remaining angular frequencies ω_{n} for n > 1 are negative. Consequently, in the gray regions, ω_{0} and ω_{1} 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 γ_{1} > 0 and γ_{2} < 0. In the stippled region, ω_{0} 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/JPOD210199.1
As in Fig. 2, but now with an upper slope ∇h_{2} = 10^{−5} in the direction 200^{°} in addition to the bottom slope in Fig. 2. The upper slope corresponds to γ_{2}/H = 0.1. The dotted line corresponds to ω_{0} and the dashed line corresponds to ω_{1}, 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 γ_{1} > 0 and γ_{2} > 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 γ_{1} < 0 and γ_{2} < 0. The two gravest angular frequencies ω_{0} and ω_{1} are both positive while the remaining angular frequencies ω_{n} for n > 1 are negative. Consequently, in the gray regions, ω_{0} and ω_{1} 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 γ_{1} > 0 and γ_{2} < 0. In the stippled region, ω_{0} 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/JPOD210199.1
As in Fig. 2, but now with an upper slope ∇h_{2} = 10^{−5} in the direction 200^{°} in addition to the bottom slope in Fig. 2. The upper slope corresponds to γ_{2}/H = 0.1. The dotted line corresponds to ω_{0} and the dashed line corresponds to ω_{1}, 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 γ_{1} > 0 and γ_{2} > 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 γ_{1} < 0 and γ_{2} < 0. The two gravest angular frequencies ω_{0} and ω_{1} are both positive while the remaining angular frequencies ω_{n} for n > 1 are negative. Consequently, in the gray regions, ω_{0} and ω_{1} 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 γ_{1} > 0 and γ_{2} < 0. In the stippled region, ω_{0} 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/JPOD210199.1
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 boundarytrapped at small horizontal scales (i.e., for kL_{d} = 7). At larger horizontal scales, we typically obtain a depthindependent mode along with another mode with largescale features in the vertical direction. Second, note that for γ_{1} and γ_{2} > 0, as in (a) and (b), the nth mode has n internal zeros, as in Sturm–Liouville theory; for γ_{1} > 0 and γ_{2} < 0, as in (c) and (d), the first two modes (n = 0, 1) have no internal zeros; and for γ_{1} and γ_{2} < 0, the zerothmode G_{0} has one internal zero, the first and second modes (G_{1} and G_{2}) have no internal zeros, and the third mode G_{2} has one internal zero. The zero crossing for the n = 0 mode in (f) is difficult to observe because the amplitude of G_{0} is small near the zero crossing.
Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPOD210199.1
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 boundarytrapped at small horizontal scales (i.e., for kL_{d} = 7). At larger horizontal scales, we typically obtain a depthindependent mode along with another mode with largescale features in the vertical direction. Second, note that for γ_{1} and γ_{2} > 0, as in (a) and (b), the nth mode has n internal zeros, as in Sturm–Liouville theory; for γ_{1} > 0 and γ_{2} < 0, as in (c) and (d), the first two modes (n = 0, 1) have no internal zeros; and for γ_{1} and γ_{2} < 0, the zerothmode G_{0} has one internal zero, the first and second modes (G_{1} and G_{2}) have no internal zeros, and the third mode G_{2} has one internal zero. The zero crossing for the n = 0 mode in (f) is difficult to observe because the amplitude of G_{0} is small near the zero crossing.
Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPOD210199.1
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 boundarytrapped at small horizontal scales (i.e., for kL_{d} = 7). At larger horizontal scales, we typically obtain a depthindependent mode along with another mode with largescale features in the vertical direction. Second, note that for γ_{1} and γ_{2} > 0, as in (a) and (b), the nth mode has n internal zeros, as in Sturm–Liouville theory; for γ_{1} > 0 and γ_{2} < 0, as in (c) and (d), the first two modes (n = 0, 1) have no internal zeros; and for γ_{1} and γ_{2} < 0, the zerothmode G_{0} has one internal zero, the first and second modes (G_{1} and G_{2}) have no internal zeros, and the third mode G_{2} has one internal zero. The zero crossing for the n = 0 mode in (f) is difficult to observe because the amplitude of G_{0} is small near the zero crossing.
Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPOD210199.1

γ_{1} > 0 and γ_{2} > 0: This corresponds to case i in section 3b. See the regions in white in Fig. 4 and Figs. 5a and 5b.

γ_{1}γ_{2} < 0: This corresponds to case ii in section 3b. See the stippled regions in Fig. 4 and Figs. 5c and 5d.

γ_{1} < 0 and γ_{2} < 0: There are two modes G_{0} and G_{1} with negative squares, 〈G_{n}, G_{n}〉 < 0, that propagate eastward and have eigenvalues G_{n} satisfying G_{n} < −k^{2} 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} > −k^{2}. The zeroth mode G_{0} has one internal zero, the first and second modes, G_{1} and G_{2}, 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
Quasigeostrophic boundary dynamics
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 boundarytrapped 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}.
Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPOD210199.1
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 boundarytrapped 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}.
Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPOD210199.1
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 boundarytrapped 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}.
Citation: Journal of Physical Oceanography 52, 2; 10.1175/JPOD210199.1
4. Eigenfunction expansions
In the following, we will only consider eigenfunction expansions that diagonalize the energy and potential enstrophy integrals of section 2e.
a. The four possible L^{2} modes

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

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

surface modes: (mixed Neumann/Dirichlet)

antisurface modes: (mixed Neumann/Dirichlet)
b. Expansions with L^{2} modes
We here examine the pointwise convergence and the termbyterm differentiability of eigenfunction expansions in terms of L^{2} modes. These properties of L^{2} Sturm–Liouville expansions may be found in Brown and Churchill (1993) and Levitan and Sargsjan (1975).^{9}
1) Pointwise equality on [z_{1}, z_{2}]
However, even though
2) Differentiability of the series expansion
Although we obtain pointwise equality on the whole interval [z_{1}, z_{2}] 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 L^{2} degrees of freedom while the surface potential vorticities, r_{1}_{k} and r_{2}_{k}, are associated with the ℂ^{2} degrees of freedom.
c. Quasigeostrophic L^{2} ⊕ ℂ^{2} modes
Smith and Vanneste (2012) investigate an equivalent eigenvalue problem to (80) and conclude that, when D_{1} and D_{2} are positive, the resulting eigenfunctions form a basis of L^{2} ⊕ ℂ^{2}. 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 L^{2} ⊕ ℂ^{2} modes
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 waveturbulence 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 L^{2} ⊕ ℂ^{2} but should also represent the vertical structure of Rossby waves in the linear (quiescent ocean) limit. Such a basis would then be amenable to waveturbulence 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 D_{1}, D_{2} > 0 that diagonalizes the energy and potential enstrophy integrals.
The Rhines modes of section 3b offer a basis of L^{2} ⊕ 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 D_{2} = 0 in the eigenvalue problem (80) and let D_{1} = γ_{1}, we then obtain the Rhines modes. Note that since D_{1} = γ_{1} = γ_{1}(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).
Galerkin approximations with L^{2} modes
Both the L^{2} baroclinic modes and the L^{2} ⊕ ℂ^{2} 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).
The two expressions Eqs. (77) and (92) are only equivalent when r_{1}_{k} = r_{2}_{k} = 0. For nonzero r_{1}_{k} and r_{2}_{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
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 L^{2} modes: the baroclinic modes, the antibaroclinic modes, the surface modes, and the antisurface modes. Additionally, we explored the properties of the family of L^{2} ⊕ ℂ^{2} bases introduced by Smith and Vanneste (2012) that contain two free parameters D_{1} and D_{2} and generalized the family to allow for D_{1}, D_{2} < 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 L^{2} and L^{2} ⊕ ℂ^{2} vertical velocity modes.
For the streamfunction L^{2} modes, only the baroclinic modes are capable of converging pointwise to any quasigeostrophic state on the interval [z_{1}, z_{2}], whereas for the vertical velocity L^{2} modes, only the antibaroclinic modes are capable. However, in both cases, the resulting eigenfunction expansion is not differentiable at the boundaries, z = z_{1}, z_{2}. Consequently, while we can recover the volume potential vorticity density q_{k}, we cannot recover the surface potential vorticity densities r_{1}_{k} and r_{2}_{k}. Thus, we lose two degrees of freedom when projecting onto the baroclinic modes. In contrast, L^{2} ⊕ ℂ^{2} modes provide an equivalent representation of the function in question. Namely, the eigenfunction expansion is differentiable on the closed interval [z_{1}, z_{2}] so that we can recover q_{k}, r_{1}_{k}, and r_{2}_{k} from the series expansion.
We have also introduced a new set of modes, the Rhines modes, that form a basis of L^{2} ⊕ 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 waveinteraction 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 L^{2}[D] is more subtle than is presented here. Namely, elements of L^{2}[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
That γ_{1} 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 z_{2} = ∞.
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
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 λDependent 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 realvalued 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).
APPENDIX B
Polarization Relations and the Vertical Velocity Eigenvalue Problem
a. Polarization relations
b. The vertical velocity eigenvalue problem
When only one boundary condition is λ dependent (e.g., γ_{2} = 0), the eigenvalue problem in Eqs. (B12) and (B13) satisfies Eq. (A4) when γ_{1} > 0 and Eqs. (A7) and (A8) when γ_{1} < 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 L^{2} modes
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