On stratified water waves with critical layers and Coriolis forces

We consider nonlinear traveling waves in a two-dimensional fluid subject to the effects of vorticity, stratification, and in-plane Coriolis forces. We first observe that the terms representing the Coriolis forces can be completely eliminated by a change of variables. This does not appear to be well-known, and helps to organize some of the existing literature. Second we give a rigorous existence result for periodic waves in a two-layer system with a free surface and constant densities and vorticities in each layer, allowing for the presence of critical layers. We augment the problem with four physically-motivated constraints, and phrase our hypotheses directly in terms of the explicit dispersion relation for the problem. This approach smooths the way for further generalizations, some of which we briefly outline at the end of the paper.


1.
Introduction. This paper concerns traveling waves in a two-dimensional inviscid and incompressible fluid lying above a flat bed. The fluid is divided into one or more layers separated by internal interfaces across which the pressure is continuous, and is bounded above either by a free surface held at constant (atmospheric) pressure or else by a rigid lid. While the velocity field is incompressible, the density of the fluid is allowed to vary continuously within each layer and discontinuously across the internal interfaces. Similarly we allow for nonzero vorticity in each layer as well as the existence of closed streamlines. Finally, we allow for Coriolis forces perpendicular to the fluid velocity. Such terms appear in non-traditional f -plane approximations at the equator [8].
Our first result (Proposition 2.1 below) is that traveling-wave solutions with Coriolis parameter Ω = 0 can be naturally associated to solutions with Ω = 0 and conversely. In this sense the two problems are mathematically equivalent, even if their physical interpretations are different. We were surprised not to find this remarked upon in recent work on waves with Coriolis forces. The basic idea is simple: By incompressibility, the Coriolis terms in the momentum equations are a gradient and so can be absorbed into the pressure. In general this redefinition of the pressure leaves forcing terms on the internal interfaces and free surface, but for traveling waves one can arrange for these forcing terms to vanish. The drawback is that the gravitational constant g must be replaced by g − 2Ωc where c is the wave This is the type of configuration which will be considered in Section 3.
speed. Branches of solutions with fixed g and variable c are therefore not preserved under this transformation. Our second result (Theorem 3.1 below) is on the existence of periodic waves. We specialize to the two-layer case with a free surface, and require the vorticity and density to be constant in each layer. We also enforce four integral constraints which ensure that the average depths of the two layers are constant, that the wave speed c is physically defined, and that the average strength of the vortex sheet at the internal interface is zero. The results are stated entirely in terms of the formal dispersion relation d(k, c) = 0 between the wavenumber k and wave speed c of an infinitesimal wave. Especially since the linear operators involved are not Fourier multipliers, it is not immediately obvious that this should be possible. We state and prove a functional-analytic lemma which clarifies the issue and allows our existence result to be more easily generalized in a variety of directions.
1.1. Governing equations. Consider a configuration with N ≥ 1 layers as in Figure 1. The layers are numbered 1, . . . , N starting with the deepest layer, while the internal interfaces are numbered 0, . . . , N with 0 corresponding to the flat bed and N to the free surface or rigid lid. Introducing a "reference thickness" h i > 0 for each layer, the "reference height" of the ith interface is h 0 + · · · + h i , and we assume that the interface itself is a graph for some function η i . On the flat bed η 0 ≡ 0. The ith layer is then where we are assuming η i−1 < h i +η i so that the interfaces do not touch. Each layer has a velocity field (u i , w i ), pressure field p i , and density field ρ i > 0, and we define the corresponding vortices by ω i = u iz − w ix . We will always work with classical solutions having at least the regularity η i ∈ C 1 and u i , w i , p i , ρ i ∈ C 1 (D i × R). For convenience we set u 0 = w 0 = 0.
In each layer D i the incompressible Euler equations hold, where here g is the acceleration due to gravity and Ω is the angular velocity responsible for the Coriolis forces. At each interface S i , including the bed S 0 and free surface S N , there are kinematic boundary conditions If the upper boundary S N is a free surface then we have there for some constant atmospheric pressure p atm . If S N is instead a rigid lid then we simply have By a traveling wave we mean a solution of (1.3) where the dependent variables u i , w i , ρ i , p i , η i depend on x and t only through the combination x−ct for some wave speed c ∈ R. Inserting this ansatz into (1.3), we are left with the time-independent problem  (1.4) in the absence of Coriolis forces, even when we leave out important work on travelingwave solutions to approximate models, on three-dimensional waves, and on the full time-dependent problem. We refer the reader to the surveys [16,17,34,39,40] and monograph [9] for a general overview. In terms of existence results for periodic waves, the simplest case of a single irrotational layer with a free surface dates back to Nekrasov [35] and Levi-Civita [30] in the 1920's. By comparison, the smallamplitude existence theory for waves with critical layers is not even a decade old [41] and the large-amplitude theory is quite recent [11]. Two-layer waves with vorticity and a rigid lid were constructed by Walsh, Bühler, and Shatah [43]; also see [29]. Like earlier work [12] with a single layer, they assume u < c throughout the fluid which rules out the existence of critical layers. Matioc [32] has subsequently given an existence theory without this assumption. Compared to [43,32], our existence result Theorem 3.1 treats the more complicated free-surface boundary condition. This introduces an additional unknown η 2 into the problem, and the dispersion relation (3.1) (for piecewise-constant vorticity) becomes quartic in the wave speed rather than quadratic. Perhaps more importantly, the dispersion relation loses monotonicity in the wavenumber k, so that there can be resonances between different wavenumbers k 1 , k 2 for fixed c. On the other hand, while [43,32] allow for general distributions of vorticity, we restrict to piecewiseconstant vorticity. Our approach is not fundamentally restricted to this choice, however; see the remarks in Section 4.3.
We also mention a recent result of Wang [44], which treats rotational waves with general vorticity and a free surface boundary condition, but does not allow for critical layers. This paper constructs not only periodic waves but also waves which are solitary (localized) and "generalized solitary" (asymptotically periodic). Unlike [43,32] which use the Crandall-Rabinowitz theorem [14] on bifurcation from a simple eigenvalue, Wang uses spatial dynamics techniques, in particular a center manifold theorem due to Mielke [33]. Other closely-related work includes [36], which allows for surface tension at the interface but requires the layers to be irrotational, and [2], which requires the density to be constant but allows a general discontinuous vorticity.
Our emphasis on the dispersion relation is similar in spirit to the work of Kozlov and Kuznetsov in [27] (also see [26]). They consider quite general rotational waves in a single constant-density layer, and treat two bifurcation problems: one where the Bernoulli constant is held fixed and the wavenumber k is varied, and another where the wavenumber k is fixed and the Bernoulli constant is varied. Our use of integral constraints is related to earlier work of Henry [19,18] on constant-density rotational waves with constant depth and Walsh [42] on continuously stratified waves.

1.2.2.
With Coriolis forces. Results on solutions of (1.4) with Ω = 0 are fewer in number and comparatively recent. There has been work on explicit Gerstner-type solutions in Lagrangian coordinates [20,23] as well as the existence of solutions via bifurcation theory [10,31,21]. There are also qualitative results on symmetry [22,1] and particle trajectories [37,25]. Several papers on Hamiltonian formulations of the time-dependent problem [5,7,6,24] also include (rather formal) discussions of solitary traveling waves.
As we will show in Proposition 2.1 below, one can in fact always set Ω = 0 in (1.4) after a simple change of variables. Thus the full strength of the classical theory for waves without Coriolis forces applies at once. In particular, some of the results for Ω = 0 mentioned in the above paragraph can be directly inferred from earlier work with Ω = 0.
1.3. Plan of the paper. In Section 2, we state and prove Proposition 2.1 on the equivalence between traveling waves with Ω = 0 and Ω = 0. For completeness we also briefly discuss a similar transformation for time-dependent waves which appears to be less useful. In Section 3, we prove our existence result Theorem 3.1. In Section 3.1, the full problem (1.4) is reduced to an abstract nonlinear equation in Banach spaces and the Crandall-Rabinowitz theorem is stated. In Section 3.2, we calculate the Fredholm indices of the relevant linear operators using standard techniques for elliptic problems. In Section 3.3, we give an abstract lemma which is useful for proving the remaining hypotheses of the Crandall-Rabinowitz theorem. In Section 3.4, we apply the lemma to complete the proof of Theorem 3.1. Finally, in Section 4 we consider several variants of Theorem 3.1, including a result where the wavenumber k is the bifurcation parameter. We have endeavored to write the paper in such a way that these variants and other generalizations are easily proved.
2.1. Traveling waves. In this section we show how the Coriolis terms involving Ω in the traveling-wave system (1.4) can be eliminated. The change of variables involves the "pseudo-stream functions" Ψ i defined up to additive constants by The existence of the Ψ i follows immediately from the identity which holds thanks to (1.4c) and (1.4d). The kinematic conditions (1.4e) and (1.4f) imply that Ψ i is constant on S i and S i+1 . Thus we can add constants to each of the Ψ i to ensure that the normalization conditions Ψ i = Ψ i+1 on S i , i = 1, . . . , N − 1, are satisfied.
Proof. Subtracting the right hand side of (1.4a) from the left hand side, the terms involving p, g, Ω are Similarly, when we subtract the right hand side from the left hand side (1.4b), the relevant terms are Since the equations (1.4c)-(1.4f) do not involve p, g, Ω, they are obviously preserved, and finally the dynamic boundary conditions (1.4g) are preserved thanks to the normalization (2.2).
Remark 2.2. Proposition 2.1 continues to hold, with the same proof, when surface tension effects are included and also when (1.4) is generalized to allow for interfaces S i that are not graphs.

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Note that the transformation (2.3) leaves everything but the pressures p i , gravitational constant g, and Coriolis parameter Ω unchanged. Thus the interfaces S i , (pseudo-) stream functions such as Ψ i , the trajectories of fluid particles, and the vorticities ω i are all preserved.

2.2.
Time-dependent waves. There does not appear to be an analogue of the transformation (2.3) which completely eliminates the Coriolis terms from (1.3). Under additional assumptions, one can, however, eliminate the Coriolis terms from the momentum equations (1.3a)-(1.3b) at the cost of adding forcing terms to the dynamic boundary conditions (1.3g) and (1.3h). While it is unclear if there are any applications, we outline such a transformation here for completeness.
Suppose, for instance, that the densities ρ i are constant in each layer. Then by (1.3d) there exist stream functions Ψ i in each layer satisfying Ψ ix = −ρ i w i , Ψ iz = ρ i u i and unique up to an additive function of time t alone. The kinematic boundary conditions (1.3e)-(1.3f) imply that on S i for i = 1, . . . , N − 1, and so we can normalize the Ψ i so that . On the top S N we can similarly arrange for With these normalizations in place, consider the transformation where we are including g → g merely to emphasize the difference with (2.3). As in the proof of Proposition 2.1, the momentum equations (1.3a)-(1.3b) are unchanged, as are (1.3d)-(1.3f). On the other hand, the boundary condition (1.3g) becomes For a rigid lid (1.3i) is unchanged, while the free surface boundary condition (1.3h) becomes 3. Existence theory. This section is devoted to an existence result for (1.4). We take N = 2 layers with a free surface condition (see Figure 1b), and seek periodic waves with a fixed horizontal wave number k = κ. Abusing notation, we henceforth identify the interfaces S 0 , S 1 , S 2 and fluid layers D 1 , D 2 with their intersections with a fundamental period {|x| < π/κ}. We assume that the vorticities ω 1 , ω 2 and densities ρ 1 , ρ 2 in each layer are constants, and define the dimensionless ratio In light of Proposition 2.1 we set Ω = 0 for simplicity, but see the discussion in Section 4.1. Introducing the shorthand The formula (3.1) can of course be formally derived in many ways; it enters into our arguments in Section 3.4 as the determinant of a certain 6 × 6 matrix.
Then there is an analytic curve of solutions to (1.4), parametrized by a small parameter ε, with the following properties.
(a) (Asymptotics) As ε → 0, we have the expansions The wave speed c is uniquely determined by the requirement Before beginning the proof, let us comment on the integral conditions (3.3)-(3.5). While the constant depth condition (3.3) is certainly natural, many authors instead fix the volume fluxes M 1 , M 2 defined in (3.8) below. This choice is not unreasonable from a physical point of view, and has some mathematical advantages. For further discussion we refer the reader to [19,18]. Condition (3.4) is a normalization for the wave speed c, sometimes called "Stokes' first definition of the wave speed". It asserts that we are working in the unique reference frame where the horizontal velocity at the bed has average value zero. Many authors, for instance [12], instead fix c and use a Bernoulli constant such as B 2 in (3.6) below as the bifurcation parameter. Condition (3.5) at the internal interface is similar; it asserts that the average jump in tangential velocity is zero. This will be true, for instance, whenever the wave is "dynamically accessible" from an initial configuration with continuous velocity. An alternative would be to instead fix another Bernoulli constant, say B 1 in (3.6) below.
3.1.1. Stream function formulation. As in Section 2, we use incompressibility to introduce stream functions in each layer, except that we drop the prefactor ρ i : Using Bernoulli's law to eliminate the pressure, standard arguments lead to the following system: Here B 1 , B 2 are Bernoulli constants, while M 1 , M 2 are the x-independent volume fluxes in each layer, Throughout the analysis we will hold ω 1 , ω 2 , r, h 1 , h 2 , κ fixed, but allow M 1 , M 2 , B 1 , B 2 and c to vary with the solution, c playing the role of bifurcation parameter. See Section 4 for related results with difference choices of parameters and constants.
3.1.2. Trivial solutions. We will perturb from the family of trivial (x-independent) solutions with η 1 , η 2 ≡ 0 and (3.9) These correspond to continuous piecewise-linear shear flows with horizontal velocity U i = Ψ iz + c; see Figure 2. Inserting into (3.6) we discover formulas for while the integral constraints (3.7) are all satisfied. Observe that, depending on the values of the various parameters, the associated relative velocities u i − c = Ψ iz may vanish at isolated values of z. These are "critical layers" where the flow reverses direction.
We write a general solution as a perturbation of the trivial solution, using lowercase letters for the perturbation variables: (3.10) 3.1.3. Flattening transformations. In the absence of the critical layers mentioned above, we could make a semi-Lagrangian change of variables originally due to Dubreil-Jacotin [15], using z as the dependent variable and Ψ i as the independent variable. Indeed this transformation was used by Wang [44] for (a generalization of) our problem. Since we want to allow for critical layers, however, we are forced to use a less elegant change of coordinates, and we instead define the new vertical variable ζ by The change of variables (x, y) → (x, ζ) maps the lower and upper fluid layers D 1 , D 2 onto the periodic strips where T κ denotes the interval [−π/κ, π/κ] with periodic boundary conditions. Similarly S 0 , S 1 , S 2 are sent to This change of variables is well-defined and piecewise smooth provided the inequalities hold so that the interface and free surface do not touch each other or the bed. Since we will be considering solutions where η 1 , η 2 are small in C 2+α , (3.12) will always hold.
For the remainder of the paper we will abuse notation and consider ψ 1 , ψ 2 as functions of (x, ζ) rather than as functions of (x, z).

3.1.4.
Linearization. Using the definitions in the previous two sections to change variables in (3.6)-(3.7) is tedious but straightforward, and we omit the calculations. Under the ever-present assumption (3.12), one obtains a system of equations for the unknown functions Φ = (ψ 1 , ψ 2 , η 1 , η 2 ) on the fixed domains Ω 1 , Ω 2 and their boundaries Γ 0 , Γ 1 , Γ 2 . The traveling-wave system (3.6) becomes while the constraints (3.7) become The functions N i appearing on the right hand sides are each rational functions of their arguments and are well-defined and analytic in the region where (3.12) holds. They are genuinely nonlinear in that This much about the N j can be deduced without writing them out explicitly; indeed the precise formulas will not be needed in this paper at all and so we omit them.

STRATIFIED WAVES WITH CORIOLIS FORCES 4757
The values of c-dependent coefficients on the left hand side of (3.13), on the other hand, are crucial: Note that the coefficients β s , β i multiply the terms with the fewest derivatives in their respective equations, while c s , c i ,c i multiply the highest order terms. Thus we expect qualitative properties such as Fredholm indices to be essentially independent of β s , β i . In the (at least formal) limit of a single homogeneous and irrotational layer, c s = c i =c i = c and β s = −β i = g.
3.1.5. Abstract formulation and the Crandall-Rabinowitz theorem. Fixing once and for all a Hölder parameter α ∈ (0, 1), we work with the Banach spaces (3.16) Here the subscript 'even' denotes evenness in the horizontal variable x; 2π/κperiodicity is already encoded in (3.11). We write elements of X as and elements of Y as . . , f 12 ).
As mentioned in the previous subsection, the system (3.13) is only well-defined when the inequalities (3.12) hold. For this reason we will restrict our attention to the open subset which contains the axis {Φ = 0}. We can then write (3.13)-(3.14) abstractly as where L(c) : X → Y is a bounded linear operator depending analytically on c and is an analytic mapping between (open subsets of) Banach spaces. One can readily check that L(c) and N preserve evenness and periodicity, at which point the above boundedness and analyticity are clear. We will prove Theorem 3.1 by applying the following analytic version of the classical Crandall-Rabinowitz theorem [14].  [3]). Let L (λ) : X → Y be a bounded linear operator between Banach spaces depending analytically on a parameter λ ∈ R, and let N : U → Y be an analytic mapping defined on an open neighborhood U of (0, λ 0 ) in X ×R which is genuinely nonlinear in that N (0, λ) = 0 and N x (0, λ) = 0 for all λ. If and not on the lower-order coefficients β s , β i . For solitary wave problems the situation is far more delicate; see for instance [4]. It is useful to split X =X × R 4 and Y =Ỹ × R 4 so that we can decompose L as the matrix operator The genuinely infinite-dimensional part of the operator is then isolated in the upperleft entry T .
Subtracting (3.20e) and (3.20d), we obtain a transmission problem for (ψ 1 , ψ 2 ) alone: (3.21) Thanks to the sign conditions (3.18), (3.21) can be solved uniquely for Ψ 1 , Ψ 2 , with the Schauder estimate where here and in what follows the constant C depends only on c s , c i ,c i but can change from line to line. This follows from theorem 16.1 and the surrounding text in [28]. We can then uniquely solve (3.20d)-(3.20e) for η 1 , η 2 , with the obvious estimate  Proof. Writing the dependence on β s , β i explicitly, we can decompose T as The first term T 0 is invertible by Lemma 3.3. Since T 1 , T 2 are compact, we deduce that T is Fredholm with index 0. Since the factors of R 4 have the same dimension in X =X × R 4 and Y =Ỹ × R 4 , the full operator L is then also Fredholm with index zero by the Fredholm bordering lemma [38].

3.
3. An abstract lemma. While the Fredholm index of L(c) only depends on the structural inequalities (3.18), the remaining hypotheses in the Crandall-Rabinowitz theorem 3.2 require more detailed information. If L(c) were a Fourier multiplier acting on a single function of a single variable, the way forward would be clear, and indeed [32,27] are able to reformulate their nonlinear problems so that this is the case. Rather than pursue similar reductions here (but now to vector-valued functions of a single variable), we treat the original operator L(c) directly, using the abstract lemma below as our primary tool. The general setting is the following. We have a family of operators L(t) : X → Y which we cannot easily express in terms of operators on finite-dimensional spaces (i.e., we can Fourier transform in x, but we are still left with inhomogeneous ODEs in ζ). This problem disappears, however, if we suitably restrict the domain and range of L(t) by considering a composition Π Z L(t)E(t) : W → Z (i.e., if we set the inhomogeneous terms in the ODEs to zero and express everything in terms of boundary data). The question is then what we can conclude about the full operators L(t) by studying the simpler operators Π Z L(t)E(t).
More precisely, suppose we have smooth families of bounded linear operators L(t) and E(t) between Banach spaces that fit into the following diagram: Letting Π Z , Π V be the projections of Y onto its factors, we require Moreover we suppose that for each ∈ N there are t-independent projections P , Q and isomorphisms I , J such that for some finite dimension n depending only on , and that these projections diag- for each fixed w ∈ W and z ∈ Z, and Q j Π Z LEP = 0, Q j Q = 0, P j P = 0 for j = . (3.25b) The following result says that certain properties of L(t) can sometimes be inferred from related properties of the n × n matrices Lemma 3.5. Suppose that for some * and t * the following hold: Then (a) ker L(t * ) = span{ξ} where ξ = EI −1 * µ; and (b) L (t * )ξ / ∈ ran L(t * ).
Note that we are neither assuming nor proving that L(t * ) is Fredholm with index 0. Also, while (i)-(ii) are more or less equivalent to (a), we do not in general expect (b) to imply (iii).
Condition (iii) in Lemma 3.5 comes from the following finite-dimensional lemma. Expanding the determinant we find Comparing with (3.28) we see that M µ ∈ ran M is equivalent to (3.27) as desired.
Proof of Lemma 3.5. First we show (a). Since t = t * throughout, we suppress dependence on it. The assumption Π V LE = 0 gives at once that Π V Lξ = 0, and hence Lξ = 0 follows from the calculation in which we have used (3.25) and (i). Conversely, suppose that x ∈ ker L. Then (3.24) implies x = Ew for some w ∈ W . By (3.25a) we can then write so that applying (3.25a) again and using (3.25b) yields By (3.25b) each term in this sum must vanish, M (I P w) = 0 for all .
Our assumption (ii) therefore implies P w = 0 for = * , while (i) gives I * P * w ∈ span{µ}. This in turn implies x = EI −1 * w ∈ span{ξ} as desired. It remains to show (b). Again L, E, L , E will always be evaluated at t = t * , and so we suppress this dependence for readability. Suppose that x ∈ X solves Lx = L ξ. We must show that (iii) does not hold. Setting ω = I −1 * µ, we calculate Differentiating the assumption Π V LE = 0, we find that Π V (LE) = 0. Applying Π V to (3.29) therefore yields x + E ω ∈ ker Π V L = ran E. Thus we can write x + E ω = Ew for some w ∈ W . We now apply J * Q * Π Z to both sides of (3.29) and compare the results. On the left hand side (3.25) implies while on the right hand side we get 3.4. Application of the lemma. We now apply Lemma 3.5 to the linear operator L(c) : X → Y appearing in our problem. We decompose Y = V × Z exactly as in (3.16), and set where the first four factors will represent the boundary values of the functions ψ 1 , ψ 2 ordered from bottom to top, i.e.
Writing elements of W as our mapping E : W → X is independent of c and defined by where ψ 1 , ψ 2 are the unique solutions of the Dirichlet problems The boundedness and injectivity of E follows from standard elliptic theory. Moreover ker Π V L = ran E by construction and so (3.24) holds.
The projections P , Q and isomorphisms I , J are defined in terms of Fourier coefficients, where ∈ N corresponds to a wavenumber k = κ. Adopting the convention we abuse notation slightly and set The hypotheses in (3.25) now follow by familiar properties of Fourier series. When = 0, the last four components of P w and Q f vanish because they are nonzero Fourier modes of constant functions. Thus the relevant dimension is n = 6 and the isomorphisms I : P W → R 6 and J : P Z → R 6 drop the last four components of their arguments: When = 0, the relevant dimension is n 0 = 10 and the isomorphisms are simply I 0 = F 0 and J 0 = F 0 .
All that is left to do to apply Lemma 3.5 is to calculate the matrices M (c) = J Q Π Z L(c)EI −1 (3.34) and to study their kernels and determinants. Fix = 0, set k = κ, and consider a generic element w = (t 1 ,t 2 ,t 3 ,t 4 ,η 1 ,η 2 ) ∈ R 6 .
Then w = I −1 w is given by and we easily check that Ew = (ψ 1 ,ψ 2 ,η 1 ,η 2 , 0, 0, 0, 0) cos(kx) ∈ X, (3.35) In particular,ψ (3.36) Applying the operator L (see the left hand side of (3.13)) and collecting terms, we find that the matrix M defined in (3.34) is For = 0, we instead take a generic element w 0 ∈ R 10 of the form and find that (3.37)

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Applying L as before we obtain the 10 × 10 matrix Proof. An explicit calculation shows that (even without ( where d(k, c) was defined in (3.1). Thanks to (3.18) and k > 0, the upper-right entry ofM is nonzero, and so its kernel is at most one-dimensional.
We can now state and prove the following more precise version of our main existence result Theorem 3.1. To verify the remaining two hypotheses, we use Lemma 3.5. From Lemma 3.7 we know that M is invertible for = 1 while M 1 has a one-dimensional kernel. Moreover by hypothesis (i) in Theorem 3.1 we have Thus all of assumptions of Lemma 3.5 are satisfied, and we conclude that ker L(c * ) = span{ξ} is one-dimensional and the transversality condition L c (c * )ξ / ∈ ran L(c * ) holds. These are precisely the remaining two hypotheses of Theorem 3.2, and so we can apply the theorem to get the existence as well as most of the stated properties of the functions (Ũ ,c).
The fact thatc (0) = 0 does not follow immediately from Theorem 3.2, but from the additional fact that our nonlinear problem (3.17) is invariant under the translation x → x + π/κ; see for instance remark 4.8 in [13]. It remains only to calculate ξ =Ũ (0) ∈ X to obtain the rest of (b). From our application of Lemma 3.5 we have ξ = EI −1 1 µ where µ = (t 1 ,t 2 ,t 3 ,t 4 ,η 1 ,η 2 ) ∈ ker M 1 . Block matrix calculations with M 1 similar to those in the proof of Lemma 3.7 show that this implies (η 1 ,η 2 ) ∈ kerM 1 . We claim that the entry (M 1 ) 22 of this matrix is nonzero. Indeed, if it were zero then we would have detM 1 = (c s c i κ csch κh 2 ) 2 = 0. Thus we can assume without loss of generality that our element of the kernel haŝ η 1 = 1 andη We then recover (t 1 ,t 2 ,t 3 ,t 4 ) from  On the other hand, the waves along the resulting bifurcation curve will have different values of the gravitational constant g, which may not be desirable in applications. Nevertheless, we can modify our proof of Theorem 3.1 so that Ω = 0 and g are both held constant. By Proposition 2.1, we can accommodate Ω = 0 simply by replacing g by g − 2Ωc in (3.13)- (3.14). This changes the nonlinear terms in unimportant ways, and affects the linear terms only through the lower-order coefficients β s , β i . Thus the Fredholm index arguments in Section 3.2 and the calculations in Section 3.4 are unaffected, except of course that g must be replaced by g − 2Ωc in the dispersion relation d(k, c) = 0. Defining we therefore have the following corollary. Then there is an analytic curve of solutions to (1.4), parametrized by a small parameter ε, and satisfying (3.2)-(3.5) except that the asymptotic expansion for η 2 is replaced by cos(κx) + O(ε 2 ).

4.2.
Wave number as the bifurcation parameter. We have chosen to keep the basic wave number κ constant and used c as a bifurcation parameter, but these roles can be reversed. To avoid having parameter-dependent domains, we switch to a scaled horizontal variablex = x/κ. This replaces the tori T κ in (3.11) with T 1 at the cost of replacing the Laplacian ∆ in (3.13) (and hence in L(κ)) with the κ-dependent operator κ 2 ∂ 2 x + ∂ 2 ζ . Of course the nonlinear terms N j are modified as well. Defining X, Y, Z, V, W as before, the extension operator E(κ) : W → X is now defined in terms of the Dirichlet problems

4.3.
Non-constant vorticity. In Theorem 3.1 our solutions are perturbations of the "trivial" stream functions (3.9) representing a piecewise-linear shear flow. Much more general shear flows can also in principle be treated. To avoid getting lost in technical issues outside the scope of the present paper, we only sketch the ideas and do not state any precise results.
4.4. The Boussinesq limit. As mentioned in the introduction, the free-surface boundary condition treated in Theorem 3.1 is more complicated than the rigid-lid condition used in [43,32], as can be appreciated by inspecting the rather complicated dispersion relation (3.1). When studying internal waves with |η 2 | |η 1 |, the rigidlid problem is often put forward as a reasonable approximation of the free-surface problem.
One systematic way to derive a rigid-lid-type model from the free-surface problem is to make a Boussinesq approximation. Here the dimensionless density ratio r = (ρ 1 − ρ 2 )/ρ 2 > 0 is used as a small parameter, while the reduced gravity g = gr is held constant. Sending r → 0 does not affect (3.6a)-(3.6f), but the dynamic boundary conditions (3.6g)-(3.6h) become 1 2 |∇Ψ 2 | 2 − 1 2 |∇Ψ 1 | 2 + g η 1 = B 1 on S 1 , η 2 = 0 on S 2 , so that in particular the free surface S 2 is flat. One can analyze the resulting nonlinear problem for (Ψ 1 , Ψ 2 , η 1 ) using the techniques of this paper; indeed the calculations are considerably simpler. However the number and nature of the boundary conditions has changed, as well as the number of unknowns, and so the spaces X, Y , etc., must all be changed. As can be guessed by sending r → 0 in (3.1) with g = gr fixed, the dispersion relation is d Bous (k, c) = 0 where d Bous (k, c) = c 2 i k coth kh 1 + coth kh 2 + c i (ω 1 − ω 2 ) − g . Define d Bous (k, c) as above, and fix c, h 1 , h 2 , ω 1 , ω 2 , g . Suppose that at some wave number κ * = 0 we have (i) (Root) d Bous (κ * , c) = 0; and (ii) (Non-critical interface) c * = ω 1 h 2 . Then there is an analytic curve of solutions of the above Boussinesq system, satisfying (3.2)-(3.5) except that η 2 ≡ 0.
For a non-rigorous study of the above Boussinesq approximation in the context of the Equatorial Undercurrent, see [45]. An interesting mathematical question is to what extent this limit can be made rigorous. For instance, can the solutions in Theorem 3.1 be constructed uniformly in a neighborhood of r = 0 with a fixed g ? Since this is a singular limit (the dynamic boundary condition (3.6h) changes type), any uniform construction will likely involve the introduction of boundary layers supported near the free surface.