1.1.Motivation

There is renewed interest to construct models for the study of the dynamics in the upper ocean (i.e., above the main thermocline, including the mixed layer) such that:

Property 1) promises to deliver a fundamental understanding of ocean processes that is difficult-if not impossible-to be attained using ocean general circulation models. Property 2) enables applying a recent flow-topology-preserving framework ²⁸ to build parametrizations ¹⁷ of unresolvable submesoscale motions and this way investigating the contribution of these to transport at resolvable scales, a topic of active research ³⁹.

1.2.Background

Back in the late 1960s and early 1970s and independently by various authors ^{20 33 42}, the rotating shallow-water model was extended by allowing for horizontal and temporal variations of the density field while keeping it as well as the velocity field independent of depth. In the simplest setting, e.g., with one active layer floating atop an abyssal layer of inert fluid, the resulting inhomogeneous-layer model enables the investigation of thermodynamic processes in the upper ocean driven by heat and freshwater fluxes across the surface. Due to the two-dimensional nature of the model, the computational cost involved in such an investigation is considerably much lower than that produced by an ocean general circulation model ^{2 37}.

Following the nomenclature introduced in ⁵¹, we will refer to the model above as IL⁰, which represents an inhomogenous-layer model wherein fields are not allowed to vary in the vertical. The homogeneous-layer shallow-water model will be called HL. Additional, more recent terminology for the IL⁰ is “thermal rotating shallow-water model” ^{66 71}, which emphasizes the ability of the to include (horizontal) gradients of temperature. The is also being called the “Ripa model” in the literature ^{15 18 19 41 47 59}, in recognition of Pedro Ripa’s contribution to its understanding ^{49-51 53 55}. We will reserve that to refer to the model generalized here, which was introduced in Ref. ⁵¹.

The assessment on the computational cost efficiency of the IL⁰ holds even when more than one active layer is considered ^{34-36,60,70,72} or when the abyssal layer is activated and rests over irregular topography ^{6 7 44}. Furthermore, due to the simplicity of the IL⁰ compared to the primitive equations for arbitrarily stratified fluid, referred to herein as IL^∞, it has facilitated conceptual understanding of basic aspects of the upper-ocean dynamics and thermodynamics ^{13 54 56 57}. Due in part to this very important reason, namely, the possibility to gain insight that is difficult to attain with an ocean general circulation model, the IL⁰ has been recently revisited ^13,24,31,70.

A multilayer version of the IL⁰ was derived in ⁴⁹, and a low-frequency approximation was developed in ⁵³; cf. recent derivations in ^64,65. The no-vertical-variation ansatz cannot be maintained under the exact dynamics produced by the IL^∞ when horizontal density gradients are present. The recipe used to keep the dynamical field’s depth-independent is to vertically average the horizontal pressure gradient. (Some authors ²² postulate a turbulent momentum flux that exactly cancels the vertical variation of the horizontal pressure gradient, but this simply is an ad-hoc hypothesis which does not contribute to the understanding of the problem.) While this is an approximation, ⁴⁹ showed that it does not spoil the integrals of motion and generalized Hamiltonian structure of the problem.

Furthermore, the IL⁰ possesses a Lie-Poisson Hamiltonian structure (18) and associated with it an Euler-Poincare variational formulation (16) wherein the Hamilton principle’s Lagrangian follows by vertically averaging that of the IL^∞29. When the equations of motion are derived in this formulation, there is a natural way to express three fundamental relations ³¹. These are 1) the Kelvin circulation theorem, 2) the advection equation for potential vorticity, and 3) an infinite family of conserved Casimir invariants (arising from Noether’s theorem for the symmetry of Eulerian fluid quantities under Lagrangian particle relabelling). The Euler-Poincare formulation provides a means to consistently introduce data-driven parameterizations of stochastic transport using the SALT (stochastic advection by Lie transport) algorithm ^{29 29}, enabling data assimilation in a geometry-preserving context.

The IL⁰ provides an attractive framework for applying the SALT algorithm to derive parameterizations for unresolved submesoscale motions in the upper ocean. Indeed, numerical simulations of the IL^{024 43 46} tend to reveal small-scale circulations that resemble quite well ⁴⁰ submesoscale filament rollups often observed in satellite-derived ocean color images. Such submesoscale motions may be unresolvable in many computational simulations. The extent to which they contribute to fluid transport at resolvable scales is a subject of active investigation ³⁹ that the SALT stochastic version of the IL⁰ may cast light on.

1.3.Limitations of the IL⁰

Despite the above geometric properties of the IL⁰, it has several less attractive aspects, which can be consequential for the production of small-scale circulations in the model. Discussed in detail by (55), these include:

An important additional limitation imposed by the depth independence of the dynamical fields, and particularly the buoyancy, is:

1.4.The IL¹

To cure the unwanted features of the IL⁰, Ref. ⁵¹ proposed the following improved closure to incorporate thermodynamic processes in a one-layer ocean model not restricted to low frequencies:

in addition to allowing arbitrary velocity and buoyancy variations in horizontal position and time, the velocity and buoyancy fields are also allowed to vary linearly with depth.

Ripa’s single-layer model, denoted IL¹, enjoys many properties which make it very promising. For instance:

3.1.Submodels

Any initial state with uniform buoyancy inside each layer (ϑ¯i=const and ϑiσ≡0) and vanishing vertical shear (uiσ≡0) is readily seen to be preserved by (12); consequently, the n-HL (a model with n homogeneous layers) follows from (12) as a particular case, just as it does it from the (exact, three-dimensional) IL^∞ model (11). In other words, the n-HL evolves on an invariant submanifold of both the n-IL¹ and IL^∞. Noteworthy, the n-HL is exact for a stepwise density stratification; however, as mentioned above, it is not able to accommodate thermodynamic processes, e.g., due to heat and buoyancy fluxes across the ocean surface. The n-IL⁰ developed in ⁴⁹ follows from (12) upon neglecting uiσ and ϑiσ; note that an initial condition with uiσ≡0 and ϑiσ≡0 is preserved neither by (12) nor by (11), so the n- IL⁰ is not a particular solution of neither the n-IL¹ nor the IL^∞. Ignoring uiσ in (12) results in a model with uiσ≡0 but ϑiσ≠0, which provides a generalization for Schopf and Cane’s [1983] intermediate layer model. Alternatively, the omission of ϑiσ in the system (12) gives a model with ϑiσ≡0 but uiσ≠0. This model differs from earlier related models ^{8 61 68} in that it is not restricted to low-frequency motions and that it explicitly represents vertical shear within each of an arbitrary number of layers.

3.2.Layer boundaries

Consistent with ansatz (6) and the assumption of zero mass transport across layer boundaries, the σ-vertical velocity (11e) in the ith layer reads

which vanishes at the base and the top of the layer. Consequently, (∂t+[u¯i∓uiσ]⋅∇)(h~i-1+(1/2)[1∓1]hi∓[ζ¯i∓ζiσ])=0 at the base of the ith layer and (∂t+[u¯i±uiσ]⋅∇)(h~i-1+(1/2)[1±1]hi∓[ζ¯i±ζiσ])=0 at the top of the ith layer. Namely, the layer boundaries (interfaces and rigid bottom or lid) of the n- are material surfaces on which each fluid particle retains its density. This includes the particular situation in which all fluid particles on these boundaries have the same density, i.e., h̃i-1+121∓1hi∓[ζ¯i∓ζiσ]=const at the base of the ith layer and h̃i-1+121±1hi∓[ζ¯i±ζiσ]=const at the top of the ith layer. The latter situation, which is most likely to happen far away from the ocean surface, cannot be described by the with only one layer.

3.3.Conservation laws

In a closed horizontal domain, on whose boundary (14) are satisfied, conservation of the ith-layer volume, mass, and buoyancy variance follows, respectively, because of (12c),

and

The total energy (sum of the energies in each layer) is also preserved in a closed horizontal domain since

where

Ei:=12hiu¯i2+16hi(uiσ)2\na+12hi2(ϑ¯i-13ϑiσ)+hih~i-1ϑ¯i, (18b)

And

b¯i:=12u¯i2+16(uiσ)2+hi(ϑ¯i-13ϑiσ)+h~i-1ϑ¯i+∑j=i+1nhjϑ¯j, (18c)

biσ:=u¯i⋅uiσ+(h~i-1+12hi)ϑiσ, (18d)

which are the mean and σ components of the ith-layer Bernoulli head. The above result follows upon realizing that ∑j=1nhjϑ¯j∂th~j-1-∑j=1n∂thj∑k=j+1nhkϑ¯k≡0, and is largely facilitated by rewriting (12d,e) in the form

∂tuiσ+hiz^×(qiσu¯i+q¯iuiσ)+∇biσ=Riσ. (19b)

Here,

is the vertical average of the ith-layer σ-vertical velocity (12);

are the mean and σ components of the ith-layer σ-potential vorticity;^a and

Riσ:=(h̃i-1+1/2hi)∇ϑiσ-1/2hi∇ϑ¯i, (22b)

which are rotational forces that arise as a consequence of the buoyancy inhomogeneities within each layer (∇ϑ¯i≠0≠∇ϑiσ).

In turn, the local conservation law for the sum of the zonal momenta within each layer is given by

where

FiM:=Miu¯i+13hiuiσuiσ+12γhi2(ϑ¯i-13ϑiσ)x^+γhi+1ϑ¯i+1∑j=1i-1hjx^ (23b)

with u_i denoting the zonal component of u_i and x^ the unit vector in the same direction ⁱⁱⁱ . The above result follows upon multiplying by γhi the zonal component of (12d),

and realizing that

∑jhjϑ¯j∂x(h~j-1-h0)+∑jhj∂x∑k=j+1nhkϑ¯k≡∂x∑jhj+1ϑ¯j+1∑k=1j-1hk.

At this point, it is crucial to specify whether the geometry is flat or spherical. On the sphere, ∇a=γ-1∂xa,∂ya, for any scalar a(x), and ∇⋅a=γ-1[∂xa+∂yγb], for any vector a = (a,b), where x=(λ-λ0)cosθ R and y=(θ-θ0)R are, respectively, rescaled geographic longitude and latitude on the surface of the Earth whose mean radius is R; and γ(y):=cosθ0cosθ and τ(y):=R-1tanθ≡-γ-1dγ/dy are coefficients that characterize the geometry of the space (the arclength element square and area element are dx2=γ2dx2+dy2 and d2x=γdxdy, respectively). The ith zonal momentum (angular momentum around the Earth’s axis) is then given by

where Ω is the Earth’s angular rotation rate. In the classical β plane, γ = 1 and τ = 0 so that all the operators are Cartesian and Mi=hi(u¯i-f0y-(1/2)βy2). However, the geometry in a consistent β plane cannot be Cartesian; instead γ=1-τ0y, τ=τ0/γ, and Mi=hi[γu¯i-f0y-12β1-R2τ02y2] ⁵⁴. Finally, conservation of the total zonal momentum (sum over all layers) in a horizontal domain in addition requires, in all cases, that both the topography and coasts be zonally symmetric.

3.4.Circulation theorems

In the IL^∞, the circulation of u+uf, where z^⋅∇×uf:=f, around a material loop, is constant in time if the latter is chosen to lie on an isopycnic surface ⁱⁱ . This is known as the Kelvin circulation theorem, which via Stokes’ theorem implies conservation of ϑ-potential vorticity. From the Hamiltonian mechanic’s side, the Kelvin theorem is the geometrical statement of invariance of the fluid action integral on level surfaces of ϑ ²⁷. The existence of a Kelvin circulation property is thus closely related to the existence of a (constrained) Hamilton’s principle for the IL^∞. The n-IL¹ does not hold such a circulation property. As a consequence, the evolution of the ith-layer ϑ -potential vorticity is not correctly represented. In Ref. ⁵¹ it is shown that this is the result of the lack of information on the vertical curvature of the horizontal velocity field. It is easy to show, however, that the evolution of the three components of the vorticity field is correctly represented, and, consistent with the IL^∞, neither q¯i nor qiσ are conserved. The evolution equations of the latter fields and the horizontal vorticity are given by (4.21) in Ref. ⁵¹ evaluated in the ith layer (note that evaluation of v in the ith layer does not simply mean replacing h by h _i). The nonexistence of a Kelvin circulation property for the n-IL¹ suggests that finding Hamilton’s principle for it is, at least, nontrivial. The n-IL¹ is nonetheless shown in Sec. 3.5 to admit a formulation suggestive of a generalized Hamiltonian structure. The n-IL⁰, surprisingly, possesses a Kelvin circulation property since

\d\dt∮lt(u¯i)(u¯i+uf)⋅dx=∮lt(u¯i)(h~i-1∇ϑ¯i+12hi∇ϑ¯i)⋅dx

holds in that model, and the material loop lt(u¯i) can be chosen to lie on an isopycnic surface. Consistent with the presence of this property, in ¹⁸ it is shown that the IL⁰ has a Lie-Poisson Hamiltonian structure which implies an analogous Euler-Poincare variational formulation ³¹ and, hence, the existence of a Lagrangian functional.

In the n-IL⁰, the following circulations theorems hold:

ddt∮∂Du¯i⋅dx=∮∂D(R¯i-μ¯iuiσ)⋅dx,

and

ddt∮∂Duiσ⋅dx=∮∂DRiσ⋅dx.

This contrasts with the IL^∞ for which the circulation of u around ∂D is time-independent. Note that the circulation of uiσ around ∂D would be invariant if both ϑ¯i and ϑiσ were chosen such that n^×∇ϑ¯i=0=n^×∇ϑiσ on ∂D ^iv . However, the latter boundary is not preserved by the n-dynamics. In opposition, the condition n^×∇ϑ¯i=0 on ∂D is preserved by the n-IL⁰ dynamics, thereby guaranteeing invariance of the circulation of u¯i around ∂D. This has been shown ⁴⁹ to have important consequences for the generalized Hamiltonian structure of the IL⁰.

3.5.A formulation suggestive of a generalized Hamiltonian structure

The Euler equations of fluid mechanics possess what is called a generalized Hamiltonian structure ⁴⁰. The IL^∞ (11), which derives from the Euler equations, is also Hamiltonian in a generalized sense ¹. A good sign of the validity of any approximate model derived from IL^∞ the is the preservation of the generalized Hamiltonian structure. This section is devoted to showing that the n-IL¹ -admits a formulation suggestive of a generalized Hamiltonian structure. A stronger statement was made in Ref. ⁵¹ for 1- IL¹.

Let φ(x,t)=(φ1(x,t),…,φ7n(x,t)) be a point on the infinite-dimensional phase space with coordinates (ϑ¯i,ϑiσ,hi,u¯i,uiσ), i = 1,…,n. Consider the relevant class, say A, of sufficiently smooth real-valued functionals of φ. For any phase functional F[φ]∈A it is further assumed that its density does not depend explicitly on t, namely, F[φ]=∫DF(φ,∇φ,…,x)d2x, and that it satisfies the boundary conditions ^v

A phase functional F[φ]∈A will be said to be admissible. Introduce then the functional

where

and E _i is the energy in the ith layer (18b); its functional derivatives are given by

The latter and the zero normal flow conditions across ∂D (14) show that H is admissible. Let now

be a skew-adjoint 7 x 7 block-diagonal matrix operator where

Ki=-(000hi-1(∙)⋅∇ϑ¯ihi-1∇⋅ϑiσ(∙)000hi-1(∙)⋅∇ϑiσ3hi-1(∙)⋅∇ϑ¯i00000-hi-1(∘)∇ϑ¯i-hi-1(∘)∇ϑiσ00hi-1\uiσ∇⋅(∙)ϑiσ∇(hi-1∘)-3hi-1(∘)∇ϑ¯i0∇(hi-1uiσ⋅∙)0). (31b)

Here, the circle (resp., bullet) in parenthesis indicates operation on a scalar (resp., two-component vector). Define further a bracket operation { ,}:A×A→A as

∀F,G[φ]∈A. Then the layer model (12) can be written in the form

which is equivalent to F˙={F,H}∀F[φ]∈A.

The bracket operator (32) satisfies {F,G}=-{G,F} (anticommutativity), {F,aG+bK}=a{F,G}+b{F,K} (bilinearity), and {FG,K}=F{G,K}+G{F,K} (Leibniz’ rule), where a, b are arbitray numbers and F,G,K[φ] are any admissible functionals. The anticomutativity property follows from the skew-adjointness of the matrix operator J [boundary terms cancel out by (26)]. The bilinearity property and Leibniz’ rule are direct consequences of the bracket’s definition.

That system (12) can be cast in the form (33) appears to suggest that the n-IL¹ is Hamiltonian in a generalized sense, with the functional H and the bracket operator { ,} being the Hamiltonian and Poisson bracket, respectively. However, the bracket (32) does not seem to qualify as Poisson since {{F,G},K}+{{G,K},F}+{{K,F},G}=0 (Jacobi’s identity) does not seem to hold.

In addition to independence of the choice of phase space coordinates, the Hamiltonian structure conveys other important properties like the direct linkage of conservation laws with symmetries via Noether’s theorem ⁶². While the n-IL¹ cannot be formally proved to be Hamiltonian, its energy, H, and -M, where M[φ]:=∫jMj is the zonal momentum of the system, do appear to be generators of t- and x-translations because of (33) and ∂xφ={M,φ}, respectively. The latter assumes that M is an admissible functional, which requires the horizontal domain to be x-symmetric since (δM/δuiσ)≡0 and (δM/δv¯i)≡0, but (δM/δu¯i)=γhi≠0. Then δHH=ε{H,H}=εH≡0 for the infinitesimal variation δHφ:=ε{φ,H}=ε∂tφ induced by H, and δMH=ε{H,M}=-εM˙=-ε∫jhjϑ¯j∂xh0≡0 iff ∂xh0≡0 for the infinitesimal variation δMφ:=ε{φ,M}=-ε∂xφ induced by M. Consequently, conservation of H and M are linked, respectively, to t- and x-symmetries of H (horizontal domain and topography in this case included).

A distinguishing feature of generalized Hamiltonian systems is the existence of Casimirs C[φ]∈A, satisfying {C,F}≡0 ∀F[φ]∈A. The Casimirs are thus integrals of motion, yet not related to (explicit) symmetries because {φ,C}≡0 ( C does not generate any transformation). The ith-layer integrals of volume, mass, and buoyancy variance are all admissible functionals that commute with any admissible function in the bracket in Eq. (32). The n-IL¹-does not seem to support additional “Casimir” invariants.

The possibility of deriving a stochastic n-using the SALT approach ²⁸ is constrained to the existence of a Kelvin circulation theorem, which is lacking for the n-IL¹. The lack of a Kelvin circulation theorem is tied to the nonexistence of a generalized Hamiltonian structure and associated Euler-Poincare variational formulation for the n-IL¹. While building parameterizations of unresolved submesoscale motions does not seem plausible using this flow-topolgy-preserving framework, investigating the contribution of the submesoscale motions to transport at mesoscales is still possible via direct numerical simulation. For this, the apparent generalized Hamiltonian formulation of the n-IL¹ can be helpful, as finite-difference schemes that preserve the conservation laws of the system might be sought using the bracket approach developed in ⁵⁸.

3.6.Arnold stability

In Ref. ⁵¹ it was shown that a state of rest (or a steady-state with at most a uniform zonal current) in the 1-IL¹ can be shown to be formally stable using Arnold’s [1965; 1966] method if and only if Eq. (9) is satisfied, i.e., if and only if the buoyancy is everywhere positive and increases (resp., decreases) with depth within a layer with the rigid bottom (resp., rigid lid). Arnold’s method for proving the stability of a steady solution of a system consists of searching for conditions that guarantee the sign-definiteness of a general invariant which is quadratic to the lowest-order in the deviation from that state; the resulting conditions are only sufficient ^{26 38}. In the n-IL¹ with n>1, however, Arnold’s method fails to provide stable conditions even for a state of rest and with no topography (h0≡0). The lowest-order (quadratic) contribution to that invariant, which can be called a “free energy” because it is defined with respect to a state of rest,

cannot be proved sign-definite when n>1. Here, Hi, gi and N_i are the ith-layer unperturbed depth, vertically averaged buoyancy, and Brunt-Väisälä frequency, respectively. Similarly, a state of rest in the n-for any n cannot be proved formally stable using Arnold’s method. Surprisingly, it is possible to prove the stability of a steady-state with a uniform zonal current in that model. But the condition of stability is not one of “static” stability like Eq. (9) as in the 1-IL¹. Contrarily, it is one of “baroclinic” stability since a uniform current in the n-IL⁰ has an implicit vertical shear through the thermal-wind balance. These results can all be inferred from ⁴⁹ and ⁵².

Nevertheless, there is at least a system, which has one IL¹-like layer and n -1 IL¹-like layers, for which a state of rest can be proved formally stable. For instance, choosing the uppermost layer to be IL⁰ -like, the corresponding free energy takes the form

where α:=n (resp., α:=1) for the rigid-bottom (resp., rigid-lid) configuration, and Hi, gi, and N _i are all constants. The above free energy is positive-definite if and only if (9) if fulfilled. [The n-HL has an infinite set of invariants which are given by ∫jhjF(q¯j) where F(⋅) is arbitrary; these include the volume integral, which is the only one needed to obtain the above result.] When all layers are homogeneous, the same result is obtained. When one IL⁰-like layer is included, however, the free energy cannot be shown of one sign.

That a steady-state (with or without a current) of the n IL⁰-cannot be proved formally stable does not mean that such a state is unstable; it means that Arnold’s method is not useful to provide sufficient conditions for the stability of that state.

3.7.Waves

The n-IL¹ Eqs. (12), linearized with respect to a reference state with no currents, can be shown to sustain the usual midlatitude and equatorial gravity and vortical waves (Poincaré, Kelvin, Rossby, Yanai, etc.) in 2n vertical normal modes. Here I shall concentrate on how well these modes are represented by considering the phase speed of (internal) long gravity waves assuming a rigid-lid setting.

The reference state is characterized by the parameter

which must be such that 0<S<1 ^{11 51}. Here, N_r is the reference Brunt-Väisälä frequency within an active layer floating on top of an inert layer; H_r is the total thickness of the active fluid layer, and gr denotes the vertically averaged reference buoyancy within the active layer. All three reference quantities are held constant. The reference buoyancy then varies linearly from gr(1+S) at the top of the active layer to gr(1-S) at the base of the active layer. In Ref. ⁵¹, it was shown that the 1-IL¹ gives the exact result for the “equivalent” barotropic or external mode phase speed of (internal) long gravity waves for all and a very good approximation to the first internal mode phase speed for all S.

Figure 3 compares, as a function of S, the phase speed as determined by the IL^∞, n-HL, n-IL⁰, and n-IL¹ for various n. The figure shows the results for the external mode (c₀), and the first (c₁) and second (c₂) internal modes. The analytical expression for the ’s phase speed for an arbitrary mode number can be found in Ref. ⁵¹; the phase speeds for the layer models are computed numerically. The solutions of the n-HL and n-IL⁰ coincide because ϑ¯i is constant for a normal mode in the n-IL¹. These models can only support n vertical normal modes. In contrast, the n-sustains vertical normal modes up to the (n + 1)th internal mode.

Figure 3 Phase speed of (internal) long gravity waves as a function of the stratification strength in a reduced-gravity reference state with no currents. 

As noted above, the 1-IL¹ result coincides with that of for the barotropic mode. To approximate well the exact solution, two HL- or IL⁰-like layers are needed. The first internal mode solution is very well approximated using two IL¹-like layers. Four HL-like layers do not provide a similar degree of approximation. The second internal mode solution is reasonably approximated with two IL¹-like layers. The distance between the exact solution and that produced using four HL-like layers is of the same order. However, in every case, the n-HL (or the n- IL⁰) overestimates the exact phase speeds.

3.8.Baroclinic instability

As one further test of the validity of the n-IL¹, the problem of baroclinic instability, particularly upper-ocean baroclinic instability, is considered here. (A subset of the results presented here appeared in ¹⁰.) The behavior in both quasigeostrophic and ageostrophic regimes is explored. The n- IL¹solutions are compared in all cases with the IL^∞ solutions. In some cases, comparisons are also made with n-HL and n- IL⁰ solutions. In the quasigeostrophic regime, analytical expressions exist for the IL^∞ solutions. Analytical or semianalytical formulas for the dispersion relations also exist in this regime for the 1-IL¹ and models with one IL⁰-like or two HL-like layers. The rest of the solutions shown are computed numerically upon finite differencing the corresponding eigenvalue problems.

Upper-ocean baroclinic instability, e.g., above the ocean thermocline, is studied in ¹¹ using the IL^∞ and the 1-IL¹ in a reduced-gravity setting. A basic state with a parallel current U=U(z)x^ is considered in that work to lie in an infinite channel on the f plane, to have a uniform vertical shear, and to be in thermal-wind balance with the across-channel buoyancy gradient. The basic velocity is further set to vary (linearly) from U¯+Uσ at the top of the active layer to U¯-Uσ at the base of the active layer. Accordingly, the basic buoyancy field Θ(y,z) varies from gr(1-2fUσy/Hr+S) at the top of the active layer to gr(1-2fUσy/Hr-S) at the base of the active layer (y is the across-channel coordinate). A nonvanishing velocity at the base of the active layer implies that the latter has a linear y-slope given by gr-1fUσ-U¯/(1-S). This basic state is a steady solution of the IL^∞ to the lowest order in the Rossby number, Ro:=U¯/L|f|∼Uσ/L|f| where L is the relevant length scale, which is assumed to be an infinitesimal parameter. In the limit of weak stratification (S→0), the horizontal scales

are well separated (RE≫RI), and thus long and short normal-mode perturbations to this state can be identified. Under long small-Rossby-number normal-mode perturbations, the base of the active layer behaves as a free boundary. For short small-Rossby-number normal-mode perturbations, this interface is effectively rigid. When the vertical shear is assumed strong, U¯/Uσ≪O(S-1), the short-perturbation limit corresponds to the classical Eady problem of baroclinic instability, in whose case solutions are insensitive to U¯/Uσ.

The left panel of Fig. 4 shows the minimum along-channel wavenumber, k, for instability as a function of U¯/Uσ in the long-perturbation and strong-shear limits (free-boundary baroclinic instability). The 1-gives the exact result for all U¯/Uσ . To provide a close approximation to this result for all U¯/Uσ with the n-HL, a fairly large n (cir. 25) is needed. Note that the 1-predicts, incorrectly, stability for U¯/Uσ<0 (the vertical shear in this model is implicit through the thermal-wind relation).