## 1. Introduction

Recent interest in upper-ocean dynamics and submesoscale turbulence has focused attention on surface geostrophic dynamics and the role of surface buoyancy variations. A main issue is the representation of active surface buoyancy by finite vertical truncations of the quasigeostrophic (QG) equations. Standard multilayer (e.g., Pedlosky 1987) and modal approximations (e.g., Flierl 1978) assume that there is no variation of buoyancy on the surfaces.

Only few attempts have being made to represent both surface active and interior dynamics in the QG equations. The pioneering work by Tulloch and Smith (2009) developed a “two-mode two-surface” model that represents the surface dynamics exactly and approximates the interior dynamics using the barotropic and first baroclinic modes. The interaction of surface and interior dynamics motivated the development of a new set of vertical modes that simultaneously diagonalize energy and a linear combination of enstrophy and surface buoyancy variance (Smith and Vanneste 2013). Other studies of the interaction of surface and interior dynamics avoid vertical modes and use instead finite-difference schemes (Roullet et al. 2012) or idealize the interior potential vorticity as a delta function sheet (Callies et al. 2015).

*p*

_{n}(

*z*), are defined by the Sturm–Liouville eigenproblem

*z*=

*z*

^{−}) and top (

*z*=

*z*

^{+}) surfaces of the domain:

*N*(

*z*) is the buoyancy frequency and

*f*

_{0}is the Coriolis parameter. The eigenvalue

*κ*

_{n}in (1) is the deformation wavenumber of the

*n*th mode. With normalization, the modes satisfy the orthogonality condition

*p*

_{0}= 1 and

*κ*

_{0}= 0.

*p*

_{n}} (Gill 1982; Pedlosky 1987; Vallis 2006; Ferrari and Wunsch 2010; LaCasce 2012). In fact, the set {

*p*

_{n}} is mathematically complete and can be used to represent any field with a finite square integral:

*f*(

*z*) has a nonzero derivative at

*z*

^{±}, or internal discontinuities, its representation as a linear combination of the basis functions {

*p*

_{n}} converges in

*L*

^{2}(

*z*

^{−},

*z*

^{+}), that is, the integral of the squared error goes to zero as the number of basis functions increases (e.g., Hunter and Nachtergaele 2001, chapter 10). In quasigeostrophic dynamics, both the streamfunction

*ψ*and the potential vorticity (PV)

*q*satisfy the requirement (4), and thus both

*ψ*and

*q*can be effectively represented by linearly combining {

*p*

_{n}}.

Despite the rigorous assurance of completeness in the previous paragraph, the utility of {*p*_{n}} for problems with nonuniform surface buoyancy has been questioned by several authors (e.g., Lapeyre 2009; Roullet et al. 2012; Smith and Vanneste 2013). These authors argue that the homogeneous boundary conditions in (2) are incompatible with nonzero surface buoyancy and that representation of the streamfunction *ψ* as a linear combination of {*p*_{n}} is useless if *ψ*_{z} is nonzero on the surfaces.

The aim of this paper is to obtain a good Galerkin approximation to solutions of the QG equation with nonzero surface buoyancy using the familiar basis {*p*_{n}}. We show that that both the inversion problem and evolutionary dynamics can be handled using {*p*_{n}} to represent the streamfunction. As part of this program we revisit and extend two existing modal approximations (Flierl 1978; Tulloch and Smith 2009) and develop a new Galerkin approximation. We discuss the relative merit of the three approximations in terms of their mathematical rigor and conservation laws and illustrate their efficiency and caveats by solving linear stability problems with nonzero surface buoyancy.

Using concrete examples, we show that the concerns expressed by earlier authors regarding the suitability of the standard modes {*p*_{n}} are overstated; even with nonzero surface buoyancy, the Galerkin expansion of the streamfunction *ψ* in terms of {*p*_{n}} converges absolutely and uniformly with no Gibbs phenomena, and the same is true for the potential vorticity *q*. A modest number of terms provide a good approximation to *ψ* and *q* throughout the domain, including on the top and bottom boundaries. In other words, the surface streamfunction can be expanded in terms of {*p*_{n}} and, with enough modes, this representation can then be used to accurately calculate the advection of nonzero surface buoyancy. In section 5, we illustrate this procedure by solving the classic Eady problem using the basis {*p*_{n}} for the streamfunction.

## 2. The exact system

In this section, we summarize the basic properties of the QG system. For a detailed derivation, see Pedlosky (1987).

### a. Formulation

*ψ*(

*x*,

*y*,

*z*,

*t*), and we use the following notation:

*ϑ*is related to the buoyancy by

*b*=

*N*

^{2}

*ϑ*/

*f*

_{0}. The QG potential vorticity (QGPV) equation is

*z*=

*z*

^{+}) and bottom (

*z*=

*z*

^{−}) are that

*w*= 0, or, equivalently,

*z*

^{+}and

*z*

^{−}, for example,

*ψ*

^{+}=

*ψ*(

*x*,

*y*,

*z*

^{+},

*t*).

### b. Quadratic conservation laws

In the absence of sources and sinks, the exact QG system has four quadratic conservation laws: energy, potential enstrophy, and surface buoyancy variance at the two surfaces (e.g., Pedlosky 1987; Vallis 2006). Throughout, we assume horizontal periodic boundary conditions.

*ρ*

_{0}

*E*, where

*ρ*

_{0}is a reference density.

If *β* = 0 then there are many quadratic potential enstrophy invariants: the volume integral of *q*^{2}*A*(*z*), with *A*(*z*) an arbitrary function of the vertical coordinate, is conserved. The choice *q*^{2} at any level

*β*destroys all these quadratic potential enstrophy conservation laws, including the conservation of potential enstrophy defined simply as the volume integral of

*q*

^{2}. Multiplying the QGPV equation [(6)] by

*q*, and integrating by parts, we obtain

*ϑ*= 0 at the ground. But the

*β*term on the right of (11) can be eliminated by cross-multiplying the QGPV equation [(6)] evaluated at the surfaces

*β*selects a uniquely conserved potential enstrophy from the infinitude of

*β*= 0 potential enstrophy conservation laws:

*β*≠ 0, the surface contributions in (12) are required to form a conserved quadratic quantity involving

*q*

^{2}. Notice that

*Z*is not sign definite. To our knowledge, the conservation law in (12) is previously unremarked.

*E*and

*Z*, the surface buoyancy variance is conserved on each surface:

*β*≠ 0, the QG model has four quadratic conservation laws:

*E*,

*Z*, and the buoyancy variance at the two surfaces.

## 3. Galerkin approximation using standard vertical modes

*N*(

*z*). The coefficients

*a*

_{0}through

_{ψ}, and thus one obtains the Galerkin approximation

^{⌣}to denote a Galerkin coefficient defined via projection of a field onto a vertical mode.

_{q}defined in analogy with (14).

*ψ*and

*q*are related by the elliptic “inversion problem”

*h*)

*p*

_{n}(

*z*) and integrating over the depth. Noting the intermediate result

_{n}is the

*n*th-mode Helmholtz operator

*ϑ*

^{±}= 0. In other words, term-by-term differentiation does not produce the correct relation [(22)] between

*ψ*and

*q*.

*q*and

*ψ*,

*δ*(

*z*−

*z*

^{±}) at the surfaces. Of course, these surface

*δ*distributions do not satisfy the

*L*

^{2}convergence condition in (4), and thus the series in (26) only converge in a distributional sense (e.g., Hunter and Nachtergaele 2001). For instance, if

*f*satisfies the

*L*

^{2}convergence condition in (4), then

### An elementary example: The Eady basic state

*N*. We use nondimensional units so that the surfaces are at

*z*

^{−}= −1 and

*z*

^{+}= 0. The standard vertical modes are

*p*

_{0}= 1 and, for

*n*≥ 1,

*κ*

_{n}=

*nπ*.

*q*= 0 and

*β*= 0. The surface buoyancies are

*ϑ*

^{±}= −

*y*.

*q*= 0 is exact:

*ψ*follows from either (16) or (22) and is

*ψ*at the boundaries, the series in (31) is absolutely and uniformly convergent on the closed interval −1 ≤

*z*≤ 0. The

^{−2}behavior of the series (31) ensures uniform convergence, for example, using the

*M*test (Hunter and Nachtergaele 2001). There are no Gibbs oscillations and a modest number of terms provide a good approximation to the base velocity

*U*(Fig. 1a). All these desirable properties are lost if we differentiate (31) with respect to

*z*.

*q*= 0, by direct differentiation of (31), one obtains

*δ*distributions on the boundaries (see Figs. 1b,c). These boundary

*δ*distributions are the Brethertonian PV sheets (Bretherton 1966). To some extent, which we investigate in section 5a, the series (32) is rescued by this Brethertonian interpretation.

Of course the correct Galerkin approximation to the Eady PV *q* = 0 is the splendidly convergent series *ψ* and *q*, even if *ψ*_{z} is nonzero on the boundaries. The problem is that differentiating the *ψ* series to produce a *q* series does not produce the Galerkin approximation to *q*.

## 4. Three approximations

In (24) we noted that the Galerkin approximations to *ψ* and *q* do not exactly satisfy the inversion relation. To address this error there are at least three different approximations one can make. The three approximations are equivalent when *ϑ*^{±} = 0. In the next three subsections, we provide a detailed description of each approximation. After testing, we recommend approximation *C* as the most reliable approximation using standard vertical modes.

### a. Approximation A

*A*uses the truncated series

*ψ*. Approximation

*A*does not use the Galerkin approximation for

*q*. Instead, the approximate PV

*q*. Instead the series

*A*the

_{n}is the Helmholtz operator in (23). Following the appendix of Flierl (1978), one can use the Galerkin projection of the nonlinear evolution equation [(6)] on the modes

_{n}to obtain

_{nms}cannot be computed exactly except in cases with simple buoyancy frequency profiles. But it suffices to compute Ξ

_{nms}to high accuracy, for example, using Gaussian quadrature.

*ϑ*

^{+}=

*ϑ*

^{−}= 0 so that the surface terms in (22) vanish and then there is no difference between

*ϑ*

^{+}and

*ϑ*

^{−}to approximation

*A*. That is, in addition to the

*ψ*series (15) at

*z*

^{±}to approximate

*ψ*

^{±}in the surface boundary conditions. This approach is not satisfactory because the resulting surface buoyancy equation [(38)] is dynamically passive, that is,

*ϑ*

^{+}and

*ϑ*

^{−}do not affect the interior evolution equations in (36).

*A*has the well-known energy conservation law

*E*in (10), the modal sum above is multiplied by the depth

*h*. Approximation

*A*also has the potential enstrophy conservation law

*A*(nor by

*B*and

*C*below). Finally, with the surface equations in (38), approximation

*A*also conserves surface buoyancy variance as in (13).

### b. Approximation B

*B*begins with the observation that the exact solution of the inversion problem in (19) and (20) can be decomposed as

*ψ*

^{I}(

*x*,

*y*,

*z*,

*t*) is the “interior streamfunction” and

*ψ*

^{S}(

*x*,

*y*,

*z*,

*t*) is the “surface streamfunction” (Lapeyre and Klein 2006; Tulloch and Smith 2009).

*ψ*

^{S}(

*x*,

*y*,

*z*,

*t*) is defined as the solution of the boundary value problem

*ψ*

^{I}(

*x*,

*y*,

*z*,

*t*) is defined as the solution of the boundary value problem

Approximation *B* assumes that one can solve the surface problem in (42) and (43) without resorting to a truncated series. For instance, with constant or exponential stratifications, one can find closed-form, exact expressions for *ψ*^{S} (Tulloch and Smith 2009; LaCasce 2012). Approximation *B* requires that the two unknown Dirichlet boundary condition functions *ψ*^{S±} = *ψ*^{S}(*z*^{±}) can be obtained efficiently from specified Neumann boundary condition functions *ϑ*^{+} and *ϑ*^{−}. The Eady problem, discussed below in section 5, is a prime example in which one can obtain this Neumann-to-Dirichlet map.

*ψ*

^{S}is in hand, the approximate streamfunction is

*B*evolution equations, we introduce the streamfunction (46) into the QGPV equation [(6)] and project onto mode

*n*to obtain

_{nms}defined in (37). Approximation B assumes that the remaining integral of (49) can be evaluated exactly. This is only possible for particular models of the

*N*(

*z*) (e.g., constant buoyancy frequency profiles). In practice, however, it may suffice to compute the integral very accurately, for example, using Gaussian quadrature.

*B*are completed with the addition of buoyancy advection at the surfaces:

Approximation *B* conserves surface buoyancy variance. But the conservation laws for energy are problematic; because *ψ*^{I} is not orthogonal to *ψ*^{S} the energy, (10) is not conserved in approximation B (K. S. Smith 2014, personal communication). These nonconservative effects are quantitatively small but are nonetheless irritating. The nonorthogonality of *ψ*^{I} and *ψ*^{S} was a motivation for development of the “surface-aware” modes by Smith and Vanneste (2013).

*β*= 0, approximation B conserves potential enstrophy:

*β*≠ 0, the analog of the exact potential enstrophy (12) is not conserved.

### c. Approximation C

Approximation *C* uses truncated Galerkin approximations *ψ* and *q*. The series *ψ* and *q*. To our knowledge, approximation *C*, correctly accounting for the surface buoyancy boundary terms, has not been previously investigated.

*C*approximates both the streamfunction and the PV by Galerkin series, the derivation of the modal equations is very straightforward compared with the calculations in appendix A of Flierl (1978); one simply substitutes the truncated Galerkin series for the streamfunction (15) and PV (17) into the QGPV equation [(6)] and then projects onto mode

*n*to obtain

_{nms}is defined in (37), and we recall the relation between

*C*, there are

*ϑ*

^{±}. The approximation

*C*evolution equations are completed by advection of the surface buoyancy:

*C*the surface buoyancy fields

*ϑ*

^{±}are not passive;

*ϑ*

^{±}are related through (53).

## 5. The Eady problem

We use classical linear stability problems with nonzero surface buoyancy to illustrate how solutions to specific problems can be constructed and to assess the relative merit and efficiency of approximations *A*, *B*, and *C*. The linear analysis does not provide the full picture of convergence of the approximate solutions. Nonetheless, in turbulence simulations forced by baroclinic instability, it is necessary (but not sufficient) to accurately capture the linear stability properties.

We use nondimensional variables so that the surfaces are at *z*^{+} = 0 and *z*^{−} = −1. The Eady exact base-state velocity is given by (30) with zero PV *q* = 0 and *β* = 0.

### a. Approximation A

While the surface fields *ϑ*^{±} are dynamically passive in approximation A, the Eady problem can still be considered because the base-state PV defined via (35) converges to *δ* distributions on the boundaries (section 3).

The base-state velocity in approximation *A* is given by the series (31) and is a good approximation to the exact base-state velocity of (30). But, according to approximation *A*, there is a nonzero interior base-state PV gradient given by the series (33). As *z* = 0 and −1. But for numerical implementation of approximation *A*, we stop short of

Figure 2 shows the growth rate of the Eady instability according to approximation *A* and compares this with the exact Eady growth rate. Approximation *A* does not do well, especially at large wavenumbers. The exact Eady growth rate has a high-wavenumber cutoff. At moderate values of *A* produces unstable “bubbles” of instability at wavenumbers greater than the high-wavenumber cutoff. The growth rates in these bubbles are comparable to the true maximum growth rate. As *κ*. These spurious high-wavenumber instabilities are due to the rapidly oscillatory interior PV gradient that supports unphysical critical layers (see Fig. 3).

Growth rate for the Eady problem as a function of the zonal wavenumber (*l* = 0) using approximations *A*, *B* (exact), and *C* with various numbers of baroclinic modes

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Growth rate for the Eady problem as a function of the zonal wavenumber (*l* = 0) using approximations *A*, *B* (exact), and *C* with various numbers of baroclinic modes

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Growth rate for the Eady problem as a function of the zonal wavenumber (*l* = 0) using approximations *A*, *B* (exact), and *C* with various numbers of baroclinic modes

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Structure of *κ* = 8 unstable mode for the Eady problem obtained using approximation *A* and *z* increases. One can see the unphysical critical layer associated with the fast-oscillating base-state PV. The critical level *z*_{c} is the depth where the unstable wave speed matches the velocity of the base state. Only the top quarter of the domain is shown.

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Structure of *κ* = 8 unstable mode for the Eady problem obtained using approximation *A* and *z* increases. One can see the unphysical critical layer associated with the fast-oscillating base-state PV. The critical level *z*_{c} is the depth where the unstable wave speed matches the velocity of the base state. Only the top quarter of the domain is shown.

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Structure of *κ* = 8 unstable mode for the Eady problem obtained using approximation *A* and *z* increases. One can see the unphysical critical layer associated with the fast-oscillating base-state PV. The critical level *z*_{c} is the depth where the unstable wave speed matches the velocity of the base state. Only the top quarter of the domain is shown.

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

### b. Approximation B: The exact solution

In approximation *B*, the zero PV in the Eady problem implies *β = 0*) are trivially satisfied; there is no interior contribution *B* solves the Eady problem exactly.

*ϑ*

^{±}:

*c*

^{B}=

*ω*

^{B}/

*k*. Using the boundary conditions (60) in (59), we obtain an eigenvalue problem

### c. Approximation C

*C*expands both the streamfunction and the PV in standard vertical modes. Thus, in the Eady problem the PV is exactly zero, as it should be:

*C*does not have the spurious critical layers that bedevil

*A*. Moreover, in approximation

*C*, the

*β*= 0) in (52) are trivially satisfied, and the inversion relationship (53) provides a simple connection between the streamfunction and the fields

*ϑ*

^{±}. The base velocity for the Eady problem in approximation

*C*is the series in (31) (the same as

*A*). From the exact shear at the boundaries we obtain the exact base-state boundary variables:

*c*

^{C}converges to the exact eigenspeed, that is,

*c*

^{C}→

*c*

^{B}as

*C*successfully captures the structure of the Eady growth rate even with modest values of

### d. Remarks on convergence

The crudest truncation (i.e., *A* and *C* (Fig. 2). With one baroclinic mode (*A* and *C* converge rapidly to the exact growth rate at wavenumbers less than about 2.2 (see Fig. 2). But surprisingly the convergence of the growth rate at the most unstable mode (*κ* ≈ 1.6) is faster in approximation *A* (~^{−4}) than in approximation *C* (~^{−2}) (see Fig. 4). However, the convergence in approximation *C* is uniform; there are no spurious high-wavenumber instabilities.

Absolute error as a function of number of baroclinic modes *κ* ≈ 1.6). The dashed line is the approximation *A* error at *κ* = 8.

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Absolute error as a function of number of baroclinic modes *κ* ≈ 1.6). The dashed line is the approximation *A* error at *κ* = 8.

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Absolute error as a function of number of baroclinic modes *κ* ≈ 1.6). The dashed line is the approximation *A* error at *κ* = 8.

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Figure 4 also shows that the approximation *A* convergence of the growth rate to zero at *κ* = 8 is slow (~^{−1}). While the growth rate does converge to zero at a fixed wavenumber, such as *κ* = 8, we conjecture that there are always faster-growing modes at larger wavenumbers.

## 6. The Green problem

To further explore the relative merit and efficiency of approximations *A*, *B*, and *C*, we study the instability properties of a system with nonzero *β*. For simplicity, we consider a problem with Eady’s base state *ψ* = −(1 + *z*)*y* on a *β* plane. This is similar to the problem originally considered by Charney (1947) and Green (1960). The major difference is that Charney considered a vertically semi-infinite domain (Charney 1947; Pedlosky 1987), while we follow Green and consider a finite-depth domain with −1 < *z* < 0.

*βy*, where

Growth rate for the Green problem with *l* = 0) using approximations *A*, *B*, and *C* with various numbers of baroclinic modes

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Growth rate for the Green problem with *l* = 0) using approximations *A*, *B*, and *C* with various numbers of baroclinic modes

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Growth rate for the Green problem with *l* = 0) using approximations *A*, *B*, and *C* with various numbers of baroclinic modes

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

The Green problem supports three classes of unstable modes, indicated in the lower-right panel (*β*; 2) the “Green modes,” which are very long slowly growing modes (Vallis 2006); and 3) the high-wavenumber “Charney modes,” which are critical layer instabilities that arise from the interaction of the surface edge wave with the interior Rossby wave that is supported by nonzero *β*.

### a. Implementation of approximation A

The base state for the Green problem is the same as in the Eady problem. In approximation *A*, the *β* term adds only a diagonal term to the Eady system [(57)] (see appendix C).

### b. Implementation of approximation B

### c. Implementation of approximation C

*B*(see appendix C).

### d. Remarks on convergence

The most crude truncation (*A* and *C*. In contrast, the *B* is qualitatively consistent with the modified Eady instabilities (see Fig. 5). With a moderate number of baroclinic modes (*A*, *B*, and *C* all resolve the modified Eady modes relatively well. At the most unstable modified Eady mode (*κ* ≈ 1.9), approximation *B* has typically the smallest error because it solves the surface problem exactly. As in the Eady problem, approximation *A* converges (~^{−4}) faster than approximations *B* and *C* (~^{−2}) at the most unstable mode, but *B* and *C* converge faster at high wavenumbers.

Approximations *A*, *B*, and *C* all converge very slowly to the high-wavenumber Charney modes (Figs. 5, 6). These modes are interior critical-layer instabilities (Pedlosky 1987), and the critical layer is confined to a small region about the steering level (i.e., the depth at which the phase speed matches the base velocity; see Fig. 7). With finite base-state shear, the critical layer is always in the interior. Thus, the problem is not that standard vertical modes are inefficient because they do not satisfy inhomogeneous boundary conditions; a low-resolution finite-difference solution also presents such bubbles in high-wavenumber growth rates (not shown). Resolution of the interior critical layer, not the surface boundary condition, is a problem for all methods at high wavenumbers. The surface-aware modes of Smith and Vanneste (2013) have similar performance to approximations *B* and *C*, but also have the same limitation—a large number of vertical modes are required to resolve interior critical layers (K. S. Smith 2015, personal communication).

Absolute error as a function of number of baroclinic modes *κ* ≈ 1.9). The dashed line is the error at *κ* = 8.

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Absolute error as a function of number of baroclinic modes *κ* ≈ 1.9). The dashed line is the error at *κ* = 8.

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Absolute error as a function of number of baroclinic modes *κ* ≈ 1.9). The dashed line is the error at *κ* = 8.

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Wave structure of the *κ* = 8 unstable mode for the Green problem with *z* increases. The potential vorticity is confined to a small region, the critical layer. The critical level *z*_{c} is the depth where the unstable wave speed matches the velocity of the base state. Only the bottom quarter of the domain is shown.

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Wave structure of the *κ* = 8 unstable mode for the Green problem with *z* increases. The potential vorticity is confined to a small region, the critical layer. The critical level *z*_{c} is the depth where the unstable wave speed matches the velocity of the base state. Only the bottom quarter of the domain is shown.

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

Wave structure of the *κ* = 8 unstable mode for the Green problem with *z* increases. The potential vorticity is confined to a small region, the critical layer. The critical level *z*_{c} is the depth where the unstable wave speed matches the velocity of the base state. Only the bottom quarter of the domain is shown.

Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-15-0073.1

For example, with *κ* = 8, approximations are qualitatively inconsistent with the high-resolution finite-difference solution. For larger values of *B* and *C* scales ~^{−3}. The growth rate for approximation *A* converges painfully slowly (~^{−1}). As in the Eady problem, at large wavenumbers, the growth rate for approximation *A* is qualitatively different from that of the finite-difference solution because of spurious instabilities associated with the rapidly oscillatory base-state PV gradient in (33).

## 7. Summary and conclusions

The Galerkin approximations *A*, *B*, and *C* are equivalent if there are no buoyancy variations at the surfaces. Thus, all three approximations are well suited for applications with zero surface buoyancy (Flierl 1978; Fu and Flierl 1980; Hua and Haidvogel 1986). But with nonzero surface buoyancy, the three approximations are fundamentally different. In particular, approximation *A*, originally introduced by Flierl (1978) in a context without surface buoyancy, obtains the approximate PV by differentiating the Galerkin series for the streamfunction, and consequently its approximate PV has violent oscillations in the interior. Approximation *B* represents the PV as a Galerkin series in standard modes and calculates the streamfunction that satisfies the exact inversion problem associated with the approximate PV (Tulloch and Smith 2009). The inversion relationship is split into surface and interior problems. Because the surface streamfunction projects onto the interior solution, the energy is not diagonalized, and consequently approximation *B* has small errors in energy conservation (K. S. Smith 2014, personal communication). The surface-aware modes of Smith and Vanneste (2013) correct this problem. Approximation *C* uses the Galerkin series for both streamfunction and PV but does not satisfy the inversion problem exactly. Nevertheless, the Galerkin series for *ψ* and *q* converge absolutely and uniformly, and approximation *C* provides a good finite truncation of the QG equations that represents surface buoyancy dynamics and also conserves energy.

With nonzero interior PV gradients, the convergence of all approximations is slow for the high-wavenumber Charney-type modes. The critical layer associated with these modes spans a very small fraction of the total depth (Fig. 7). To accurately resolve these near singularities at the steering level, there is no better solution than having high vertical resolution in the interior.

For problems with nonuniform surface buoyancy and nonzero interior PV gradient, we recommend approximation C for obtaining solutions to the three-dimensional QG equations using standard vertical modes.

The codes that produced the numerical results of this paper, plotting scripts, and supplementary figures are openly available online (at https://github.com/crocha700/qg_vertical_modes).

## Acknowledgments

CBR is grateful for a helpful conversation with G. R. Flierl. We thank Joe LaCasce and Shafer Smith for reviewing this paper. We had useful discussions with Shafer Smith, regarding approximation *B* and the surface-aware modes, and with Geoff Vallis, who pointed out that Green (1960) first considered the Eady+*β* problem. This research was supported by the National Science Foundation under Award OCE 1357047.

## APPENDIX A

### Convergence of Galerkin Series in Standard Modes

*Z*∈ [0,

*π*] with boundary conditions

*γ*

_{0}and

*γ*

_{π}are real constants of arbitrary sign, and

*dS*/

*dz*= 0 at a boundary then the appropriate condition at that boundary is

*P*

_{n}= 0.

A special case of Theorem I from Jackson (1914) states that the expansion of a function *f*(*z*) as a series in eigenfunctions *P*_{n} converges absolutely and uniformly, provided that both *df*/*dz* and *d*Λ/*dz* are continuous and bounded, regardless of whether or not *f* satisfies the same boundary conditions as *P*_{n}. (The remainder of the theorem concerns the rate of convergence under stronger conditions on *ψ* and Λ.) The streamfunction, potential vorticity, and buoyancy profiles are typically assumed to be smooth in studies of QG dynamics, which implies that both *f* and Λ will satisfy the above conditions. Uniform convergence over *Z* ∈ [0, *π*] implies uniform convergence over *z* ∈ [*z*^{−}, *z*^{+}].

## APPENDIX B

### Derivation of Conservation Laws for Approximation *C*

*C*, we multiply the modal equations in (52) by

*n*to obtain

## APPENDIX C

### Details of the Stability Problems

#### a. The interaction tensor

_{ijk}is fully symmetric. Permuting the indices so that

*i*≥

*j*≥

*k*, we obtain

#### b. Approximation A

_{nms}, and the inversion relation [(35)], we rewrite row

*n*+ 1 of the linear Green system:

*n*th-mode Helmholtz operator in Fourier space is

*c*

^{A}.

#### c. Approximation B

*n*+ 1)th row originates from the

*n*th interior equation [(69)]:

*γ*_{ms}is

#### d. Approximation C

##### 1) The Eady problem

*C*become exact, that is,

## REFERENCES

Bretherton, F., 1966: Critical layer instability in baroclinic flows.

,*Quart. J. Roy. Meteor. Soc.***92**, 325–334, doi:10.1002/qj.49709239302.Callies, J., G. Flierl, R. Ferrari, and B. Fox-Kemper, 2015: The role of mixed layer instabilities in submesoscale turbulence.

, in press.*J. Fluid Mech.*Charney, J. G., 1947: The dynamics of long waves in a baroclinic westerly current.

,*J. Meteor.***4**, 136–162, doi:10.1175/1520-0469(1947)004<0136:TDOLWI>2.0.CO;2.Charney, J. G., 1971: Geostrophic turbulence.

,*J. Atmos. Sci.***28**, 1087–1095, doi:10.1175/1520-0469(1971)028<1087:GT>2.0.CO;2.Ferrari, R., and C. Wunsch, 2010: The distribution of eddy kinetic and potential energies in the global ocean.

,*Tellus***62A**, 92–108, doi:10.3402/tellusa.v62i2.15680.Flierl, G. R., 1978: Models of vertical structure and the calibration of two-layer models.

,*Dyn. Atmos. Oceans***2**, 341–381, doi:10.1016/0377-0265(78)90002-7.Fu, L.-L., and G. R. Flierl, 1980: Nonlinear energy and enstrophy transfers in a realistically stratified ocean.

,*Dyn. Atmos. Oceans***4**, 219–246, doi:10.1016/0377-0265(80)90029-9.Gill, A. E., 1982:

*Atmosphere–Ocean Dynamics.*Academic Press, 662 pp.Green, J., 1960: A problem in baroclinic stability.

,*Quart. J. Roy. Meteor. Soc.***86**, 237–251, doi:10.1002/qj.49708636813.Hua, B., and D. Haidvogel, 1986: Numerical simulations of the vertical structure of quasi-geostrophic turbulence.

,*J. Atmos. Sci.***43**, 2923–2936, doi:10.1175/1520-0469(1986)043<2923:NSOTVS>2.0.CO;2.Hunter, J. K., and B. Nachtergaele, 2001:

*Applied Analysis*. World Scientific, 456 pp.Jackson, D., 1914: On the degree of convergence of Sturm-Liouville series.

,*Trans. Amer. Math. Soc.***15**, 439–466, doi:10.1090/S0002-9947-1914-1500989-4.LaCasce, J., 2012: Surface quasigeostrophic solutions and baroclinic modes with exponential stratification.

,*J. Phys. Oceanogr.***42**, 569–580, doi:10.1175/JPO-D-11-0111.1.Lapeyre, G., 2009: What vertical mode does the altimeter reflect? On the decomposition in baroclinic modes and on a surface-trapped mode.

,*J. Phys. Oceanogr.***39**, 2857–2874, doi:10.1175/2009JPO3968.1.Lapeyre, G., and P. Klein, 2006: Dynamics of the upper oceanic layers in terms of surface quasigeostrophy theory.

,*J. Phys. Oceanogr.***36**, 165–176, doi:10.1175/JPO2840.1.Pedlosky, J., 1987:

*Geophysical Fluid Dynamics.*2nd ed. Springer-Verlag, 710 pp.Roullet, G., J. McWilliams, X. Capet, and M. Molemaker, 2012: Properties of steady geostrophic turbulence with isopycnal outcropping.

,*J. Phys. Oceanogr.***42**, 18–38, doi:10.1175/JPO-D-11-09.1.Smith, K. S., and J. Vanneste, 2013: A surface-aware projection basis for quasigeostrophic flow.

,*J. Phys. Oceanogr.***43**, 548–562, doi:10.1175/JPO-D-12-0107.1.Tulloch, R., and K. S. Smith, 2009: Quasigeostrophic turbulence with explicit surface dynamics: Application to the atmospheric energy spectrum.

,*J. Atmos. Sci.***66**, 450–467, doi:10.1175/2008JAS2653.1.Vallis, G. K., 2006:

*Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation*. Cambridge University Press, 769 pp.