A variational principle for three-dimensional water waves over Beltrami flows
We consider steady three-dimensional gravity-capillary water waves with vorticity propagating on water of finite depth. We prove a variational principle for doubly periodic waves with relative velocities given by Beltrami vector fields, under general assumptions on the wave profile.
This paper is concerned with three-dimensional steady water waves driven by gravity and surface tension. Almost all previous investigations of such waves have worked under the assumption of irrotational flow. In this paper, on the other hand, we allow for non-zero vorticity. This could be important for modelling three-dimensional interactions of waves with non-uniform currents. While our study is limited to Beltrami fields, even this particular case is a step forward compared to the previous state of knowledge. The fluid domain is assumed to be an open, simply connected set, bounded from below by a rigid flat bottom and from above by a free surface , separating the fluid from the air. Let be the (relative) velocity field and the pressure. In a moving frame of reference, the fluid motion is governed by the steady Euler equations
|with kinematic boundary condition on the top and bottom boundaries|
|and dynamic boundary condition on the free surface|
Here and is the mean curvature of the free surface defined by where is the unit outward normal, while is the surface tension coefficient.
In the classical situation the free surface is given by the graph of a function, which excludes overhanging wave profiles. In this paper we consider a more general geometry as e.g in [1, 6, 10], by defining the fluid domain as
for some and a map satisfying the following conditions:
is a diffeomorphism with bounded partial derivatives and ;
for all .
Thus, the free surface is given by
while the flat bottom is
In what follows we will use the notation
for the surface parametrization. Our assumptions allow overhang but exclude self-intersection and they imply that lies above
The set of all satisfying (F1) and (F2) will be denoted by . Here , with and , denotes the class of times continuously differentiable functions whose partial derivatives of order less than or equal to are bounded and uniformly -Hölder continuous. The notation will be used for the space of functions satisfying this condition in a neighbourhood of each point in . Throughout the rest of the paper, we will continuously extend functions in to the boundary of without explicit mention.
We will consider doubly-periodic waves as follows. Let
be a two-dimensional latticegenerated by two linearly independent vectors , and let
be a two-dimensional periodic cell in the lattice. We will assume that is periodic with respect to the lattice, so that
for all and , and we denote by the set of all satisfying this property. We will consider periodic solutions, meaning that
for all .
In the irrotational case, when everywhere in , there are several existence results for different types of three-dimensional waves, including doubly-periodic waves, fully localized solitary waves and waves with a solitary-wave profile in one horizontal direction and periodic or quasi-periodic profile in another (see e.g. [7, 8, 13, 20, 21, 22, 25, 26, 31] and references therein).
On the the other hand, the existence of genuinely three-dimensional water waves with vorticity is completely open, except for a non-existence result for water waves with constant vorticity . Even in the absence of a free surface, the literature concerning steady rotational flows with vorticity is pretty scarce. There are only a handful of general existence results for steady flows with vorticity in fixed domains [2, 9, 32]. However, the special case when the velocity and vorticity fields are collinear, that is,
for some scalar function , has received more attention. Such vector fields are known as Beltrami vector fields or force-free fields and are well-known in solar and plasma physics (see e.g. [18, 30]). Any divergence-free Beltrami field generates a solution to the Euler equation (1.1a) with pressure given by
In general, condition (1.1b) is satisfied if is constant along the streamlines of . In this paper we will however concentrate on the case when is constant throughout the whole fluid. Such fields are often called strong Beltrami fields or linear force-free fields. The theory for strong Beltrami fields is much more developed than for Beltrami fields with variable (see e.g. the discussions in  and ) and in fact an obstruction to finding fields with variable was recently discovered in . In the following, we shall simply take Beltrami fields to mean strong Beltrami fields. Beltrami fields are intimately connected with chaotic motion, the famous ABC flow  being a classical example. In  it was shown that any locally finite link can be obtained as a collection of streamlines of some Beltrami field and in  a similar result was shown for vortex tubes. Note that linear dependence of and is in some sense necessary for chaotic behaviour by a theorem of Arnold .
For a Beltrami field the governing equations are
The aim of the present paper is to find a variational formulation (Theorem 3.1) for this problem in the periodic case. Classical and modern variational formulations [12, 29, 33] have proved useful in a variety of existence and stability theories for periodic and solitary travelling water waves. This includes two- and three-dimensional waves in the irrotational setting (see the references mentioned above) as well as two-dimensional waves with vorticity [6, 10, 23, 24]. It is therefore natural to expect that a variational principle for three-dimensional waves over Beltrami flows could be useful. In the absence of a free boundary, there is a classical variational formulation by Woltjer  which was further developed by Laurence & Avellaneda  (see also the related formulation by Chandrasekhar & Woltjer ). It states that Beltrami fields are critical points of the energy subject to the constraint of fixed helicity. The presence of the free boundary requires some nontrivial modifications of this formulation, as does the different geometric setting. The first step is to construct vector potentials satisfying certain boundary conditions (Theorem 2.1). The variational principle (Theorem 3.1) is then formulated in terms of such potentials. In our presentation we have striven for a balance between rigor and simplicity. The variational formulation is presented in a mathematically rigorous fashion in terms of certain function spaces, but we have tried not to overemphasize technical details. The choice of variational formulation in Theorem 3.1 is certainly not unique. We give some comments about this after the proof of the theorem, which could be useful for a variational existence theory. In addition, it would also have been possible to use other function spaces, such as Sobolev spaces.
2 Vector potentials
A vector potential of is a vector field such that
Such a potential is not unique since we can add to it the gradient of any smooth function . In order to derive a variational principle for Beltrami flows, we need to examine the structure of vector potentials for periodic vector fields satisfying on . For and we put
The subscript per stands for periodicity with respect to the lattice . We will also need the divergence-free analogues of the above spaces, defined by
When has zero fluxes, there is a unique vector potential from , . This is no longer true for non-zero fluxes. However, one can prove the following statement.
Let us explain the connection between the constants and the fluxes corresponding to the vertical sides of a basic periodic cell of the fluid domain that are parallel to the lattice vectors and respectively. Using Stokes theorem, we find that
Here the vertical components of the contour integral cancel due to the periodicity and the top part is zero because of (2.1).
The proof of the theorem relies on the following regularity result for the Biot-Savart integral
Let be a bounded domain with -smooth boundary and let . Then and
where the constant depends only on the domain and .
Note that it is enough to consider the scalar operators
which are the partial derivatives of the Newtonian potential
see e.g. [19, Lemma 4.1]. It is well known that , which implies that
This finishes the proof of the lemma. ∎
Proof of Theorem 2.1.
Let us define
which splits the domain into simple periodic cells. Furthermore, let us consider a -domain such that . Thus, intersects only eight neighbouring cells. Now let be a partition of unity on such that (i) on ; (ii) on , where ; (iii) , . Then, for a given vector field , we let
which is the Biot-Savart integral of . The latter integral converges for all and , as can be seen using Lemma 2.2.
Because the vector fields are not periodic in general, we define
for some bounded function , the series in (2.3) converges uniformly and so is well defined and continuous everywhere. Furthermore, since
for all and , we obtain that is periodic, and . Let us prove that there exists a function such that satisfies the boundary conditions. We let
be the tangential part of the field on the top boundary. Consider the tangent vectors and and define a two-dimensional vector field by
We claim that the field is conservative, that is there exists a function (not necessarily periodic) such that . Indeed, this follows from the relation
(see [5, §97]) and the fact that on . Because is periodic, we necessarily have
where is periodic and are constants.
A similar argument is valid for the bottom boundary. In this case the tangential vectors are
and the corresponding two-dimensional field is given by
Just as before, we obtain
for some periodic function and constants .
In order to eliminate the tangential periodic part of on the boundary, we solve the Dirichlet problem
Because both functions and are periodic, there is a unique periodic solution . We also define and let be its periodic part. Letting , , be the unique solution of the Dirichlet problem
we find that on . Similarly, , while on , and , on (note that on for ). Now we put
A direct calculation shows that
on , so that
On the other hand, we get
on . Thus,
This finishes the proof of the theorem. ∎
3 Variational principle
In this section we will formulate and prove a variational principle for Beltrami vector fields with a free surface. The principle is formulated in terms of the vector potential introduced in the previous section and the domain . A key difficulty is that the surface of the domain is not fixed and is a part of the variation. The admissible domains will be parametrized by maps . One of the issues is that the vector potential depends on through its domain of definition. This can be solved by extending to the whole of . However, and are still coupled through the boundary condition imposed on the free surface. Thus, the proper way to think about the domain of the involved functionals is as a submanifold of . Rather than making this approach completely rigorous, we shall simply consider critical points along admissible families of curves. After presenting the the theorem and its proof, we will discuss some alternative perspectives on the variational formulation which might be useful for further studies.
Let be a single cell of the periodic domain defined in Section 2 for which the top boundary is given by
Let us consider the functionals
for some fixed constants . We look for critical points of the functional
where and are fixed constants. This is, at least formally, equivalent to considering critical points of subject to the constraints of fixed and . When taking variations of the functionals, we consider a family of domains , , where is family of domain parametrizations which is continuously differentiable in in the topology. Note that we have written rather than in the arguments of the functionals , and to emphasize that they only depend on and not on the specific parametrization. However, when taking derivatives later, we will write to indicate the direction in which we differentiate. We also consider a continuous family of vector fields , , such that is differentiable when considered as a map from into . The reason for only assuming differentiability with respect to the topology and not the topology is that we use compositions to construct suitable curves and that this leads to a loss of derivatives. The vector fields are assumed to satisfy conditions (3.1) and (3.2). Such curves will be called admissible as will the corresponding variations
It follows from the admissibility conditions that
along the top boundary and
on the bottom , where , etc. Note that (3.3) gives a relation between and , so that the variations are not independent.
Our main theorem is the following variational principle.
For the proof we will need the following technical lemma.
Let be given and extend it to a function with for . The function
then clearly satisfies
However, is only in . Convolving with a smooth mollifier which is -periodic and has compact support in , we obtain a function with compact support in and . We set and notice that is a diffeomorphism from to itself for sufficiently small .
Next, for a given vector field and , we define a vector field by
where . The map
is a linear homeomorphism of Banach spaces. Furthermore, satisfies the boundary conditions
belongs to . Setting
we find that is an admissible curve. ∎
Proof of Theorem 3.1.
A critical point is subject to the Euler-Lagrange equation
Let us calculate all the variations in (3.7). A direct calculation gives
where and we have used the notation for convenience. This notation will also be used for other functions below. An application of the divergence theorem to the first integral leads to
The boundary integral above equals
The boundary integral over is zero in view of (3.4). Similarly, we find
Note that any pair , with is admissible since we can choose and . Evaluating for such variations we get
Let us calculate . For this purpose we use (3.3) to write
The integrals and can be rewritten as
Note that the normal component of does not contribute to ; therefore only the tangential component is of interest. Let us show that
is the surface gradient of , and
are the dual vectors of , in the tangent plane to . Here the notation () should be interpreted as a directional derivative in the direction (). Using the identities
we can compute
On the other hand, differentiating the identity , we find that
Combining these identities with the relation
Using the identity
The second integral here is zero because of (3.1). It follows that
where is the tangential part of . Differentiating (3.1) in a tangential direction , we obtain