Far-field perturbations of vortex patches

Abstract

In this paper I investigate the dynamics of vortex patches in the Yudovitch phase space. I derive an approximation for the evolution of the vorticity in the case of nested vortex patches with distant boundaries, and study its long-time behaviour.

1. Introduction

The long-time behaviour of solutions of two-dimensional incompressible Euler equations is an interesting and highly non-trivial subject. It is well known that smooth and localized initial data lead to a global-in-time well-posed evolution in spaces of smooth functions. Beyond this, little is known about the long-time dynamics. In this paper, I consider the evolution of non-smooth solutions, in a well-known phase space of functions with a limited degree of non-smoothness. The equations of ideal incompressible fluids in two dimensions can be described in terms of a single scalar field, the vorticity ω, which is a function of space and time, ω=ω(x,t), with and . The vorticity is transported by a flow it creates: it is an active scalar. The transport

1.1

is done by an incompressible velocity field u(x,t) whose curl is the vorticity, ω=∂₁u₂−∂₂u₁. This linear relation can be inverted by writing u=∇^⊥ψ and seeking ψ whose gradient decays at infinity and solves Δψ=ω. The global existence and uniqueness of solutions of (1.1) for vorticity in the class is a classical result of Yudovitch [1]. The evolution (1.1) results in a rearrangement of the vorticity distribution by a volume-preserving transformation with quasi-Lipschitz classical trajectories. If the initial datum ω(x,0) is a step function, then it remains a step function, with only the plane domains of constant value evolving in time. An equation of evolution for the boundary of such a domain, termed ‘contour dynamics’, was derived and studied numerically by Zabusky et al. [2]. If the initial vorticity equals a constant Ω in a simply connected bounded domain with smooth boundary (a vortex patch), then evolution of vorticity is reduced to a non-local evolution equation for a complex-valued function z(α,t) representing the boundary of the vortex patch at time t, parametrized by a parameter α∈[0,2π],

1.2

The derivative z′(α,t)=∂z(α,t)/∂α obeys

1.3

This equation resembles very much the simple equation ∂_tω=ωHω [3], where H is the Hilbert transform, an equation that served as a one-dimensional scalar model for the three-dimensional vectorial vortex stretching equation. The simple equation blows up in finite time. Motivated by this, it was conjectured [4] that the vortex patch equation develops singularities in z′. It turned out [5,6] that the boundaries of vortex patches remain smooth, if they were initially so. If the initial patch is an ellipse, then it remains an ellipse for all time, and the evolution consists of a rigid rotation with constant angular velocity, around a fixed centre, the symmetry centre of vorticity. The stability of these Kirchhoff ellipses under strain or local perturbations was investigated [7,8]. In this paper, we study the effect of far-field perturbations. We derive equations for a couple of contours which approximate the Eulerian evolution when one contour is far from the other. The system becomes almost uncoupled: the outer curve has a self-determined evolution influenced by the inner curve only via a constant coefficient computed from the area of the region surrounded by the inner curve. That area is conserved under the evolution. The effect that the evolving inner curve has on the outer curve is one of pure rotation around the conserved vorticity field centre. The rotation, however, is not rigid: its angular velocity depends on radius, is constant at fixed radius, but decreases with increased radius. The evolution of the inner curve is influenced by the outer curve via a time-dependent complex coefficient ζ(t). Remarkably, if the inner curve is an ellipse, it stays an ellipse. The nonlinear stability of this ellipse is determined by the long-time correlation of ζ(t) with a geometric quantity representing the inner ellipse. If the outer curve is initially an ellipse, it does not stay one, except in the case it was a circle. If the outer curve is initially an ellipse of small eccentricity, then its evolution can be approximated for long time by that of an ellipse, and in that case ζ can be computed explicitly. The resulting system can be investigated in detail and instability can be proved. The instability is strong, in the sense that the perturbed ellipse's aspect ratio degenerates, while keeping constant area. The proof of this instability is done by studying the dynamics of a complex quantity that represents the aspect ratio of the inner ellipse and the angle it makes with a coordinate system. Degenerate ellipses are represented by the boundary of the unit disc, and the dynamics is such that there can be a stable fixed point on the boundary of the unit disc which attracts trajectories from inside the circle. This means that non-degenerate ellipses degenerate in infinite time.

2. Vortex patches

We consider the evolution of a two-dimensional incompressible inviscid fluid. We describe first the vorticity distribution. We take N smooth, disjoint, oriented, closed plane curves, , j=1,…,N. The complement of their union is an open set . We denote by D_j the connected components of D, . We denote by D_N+1 the unbounded connected component. Each curve Γ_j divides into two connected open sets. We denote the bounded one U_j. We orient Γ_j such that the vectors (n_j,τ_j), where n_j is the outer normal to U_j and τ_j is tangent to Γ_j, define the same orientation in as the standard basis (e₁,e₂). This is the same as saying that an observer travelling on Γ_j in the sense of the parametrization has U_j on his left side, or that e^iπ/2n_j=τ_j. (We identify with .) We consider the vorticity

2.1

with , k=1,…,N, Ω_N+1=0 and χ_Dk the characteristic (indicator) function of D_k. As , it is well known [1] that the incompressible Euler equations with initial data like in (2.1) possess global unique weak solutions in Y . Moreover, the solution is given implicitly by

2.2

with D_k(t) obtained from D_k(0) by the Lagrangian transformation

2.3

where

2.4

and u is the velocity vector u=∇^⊥ψ, with ∇^⊥=e^iπ/2∇,

i.e.

2.5

with

2.6

and boundary condition ∇ψ→0, as , that is,

2.7

As e^iπ/2n_k=τ_k, we obtain by the divergence theorem

2.8

where ω_k are the numbers

2.9

i.e. ω_k is the jump in ω(⋅,t) as we cross from U_k to . Because each Γ_k intersects exactly two sets , there is no ambiguity in the definition. If Γ_k is parametrized by z_k(s), with s∈[0,2π] and z_k(0)=z_k(2π), , then the integrals in (2.8) are

The velocity field defined by (2.8) is Hölder continuous, and, in particular, (2.8) is well defined for x∈Γ_j. The vortex patch equations are the equations of evolution of the curves Γ_j. If

2.10

and

2.11

then the vortex patch equations are

2.12

i.e.

2.13

The centre of the vorticity field is defined by

We check that x is conserved during the motion:

by incompressibility (). Then

Now

and thus

The expression

is antisymmetric in k and l, and thus dx/dt=0. We note that, if ψ∈C², then

2.14

and the integral because u decays like |y|⁻¹. This argument requires though the compactly supported ω to be smoother than a vortex patch (C^α suffices). If the configuration of the Γ_k is a collection of concentric ellipses (in the geometric sense), then (0,0), the centre of the vorticity field, coincides with the geometric centre.

Let us observe that, for any vorticity in the Yudovitch class , we have that

2.15

Indeed, this is easily verified by writing first

then splitting the integral in two pieces, one for |x−y|≤R and one for |x−y|≥R, and then optimizing in R. Note that if ω solves the Euler equations, then the right-hand side of (2.15) is time independent. On the other hand, it is easy to see that a velocity given by (2.10) is bounded by

2.16

where |Γ| is the length of the curve Γ. Indeed, parametrizing

with s∈[0,|Γ|] and r(0)=r(|Γ|), θ(0)=θ(|Γ|), we have, denoting d/ds by ′ and integrating by parts twice,

The inequality (2.16) follows because |r′|≤|ζ′|.

3. Elliptical vortex patches

An ellipse centred at the origin of Cartesian coordinates in the plane can be represented as

3.1

with , α∈[0,2π]. If we write z_j=r_j e^iθ_j, then the ellipse is

3.2

with

3.3

Thus, (θ₁−θ₂)/2 is a phase shift, which of course is a redundant parameter, (θ₁+θ₂)/2 represents the angle the ellipse makes with the coordinate system, and a and b are major and minor semi-axes. The convention |z₁|≥|z₂| corresponds to a choice of positive trigonometric orientation (anticlockwise). A Kirchhoff ellipse is a solution of the two-dimensional incompressible Euler equations whose vorticity is a non-zero constant Ω in a region bounded by an ellipse, and zero outside that region. The parametric representation of a Kirchhoff ellipse is [9]

3.4

with

3.5

where A is the area of the ellipse

3.6

The Kirchhoff ellipse has time-independent |z₁| and |z₂|, and therefore constant length of its semi-axes, and constant area. It rotates rigidly with angular velocity

3.7

4. Far-field perturbations of vortex patches

Let us consider a base vorticity

4.1

and a perturbed vorticity

4.2

Because, by definition D₂∩D₁=∅, we have

where |D₂| is the area of D₂. The boundaries Γ₁ and Γ₂ are described by functions z₁(α,t) and z₂(α,t) satisfying the vortex patch equations. We assume that z₂ is situated far away,

4.3

with ϵ>0 very small. The fact that Γ₂ is far from Γ₁ does not stop η from being a small perturbation in Y of ω. The vortex patch system is

4.4

and

4.5

with ω₁ of order one and ω₂ very small. Let us write, in (4.4),

and, in (4.5),

The system (4.4) and (4.5) is thus

4.6

where

4.7

and

4.8

Now we use the assumption that any |z₂| is much larger than any |z₁| and approximate the system by

4.9

where

4.10

and U₁ is given in (4.7). Let us make a few observations regarding quantities in (4.9). First,

where ind(0,Γ₂) is the index (winding number) of Γ₂ at zero. Second,

4.11

where A_j(t) is the normalized area of the region U_j bounded by the curve Γ_j:

4.12

Collecting these observations, the system (4.9) becomes

4.13

Now we claim that solutions of (4.13) have constant normalized areas A_j. Indeed,

The terms I_jj(α,t) in the integrals cancel because they lead to integrals

which are zero because of the antisymmetry of the integrand in (α,β). The rest of the terms cancel because they are integrals of derivatives of periodic functions:

and

Note the effect of the separation of Γ₂ from Γ₁: the equation for Γ₂ decouples,

4.14

where A₁ is a constant, determined once and for all from the area enclosed by the initial curve Γ₁. On the other hand, z₂ influences the evolution of z₁ only through constant (in α) terms, U₁(t), given in (4.7), the winding number around zero of Γ₂, ind(0,Γ₂) and

4.15

and

4.16

The same decoupling occurs if we have a system of N widely separated curves where the vorticity jumps from one constant value to another. Now we are going to restrict our attention to the case in which the curve z₂ has antipodal reflection symmetry

4.17

It is easy to see that, if the initial curve z₂(⋅,0) has antipodal reflection symmetry, then the solution of (4.14) has antipodal reflection symmetry for all time. This follows because the derivative also has antipodal reflection symmetry. If a curve z has antipodal reflection symmetry then

In particular, if z₂ has antipodal reflection symmetry then

4.18

If the winding number of the outer curve around the origin is 1, then the equation for z₁ becomes

4.19

and this equation respects antipodal symmetry: if initially present, the symmetry persists as long as the solution is smooth. We note that, if the winding number of the outer curve is non-zero, then, under our assumption of separation of contours, the outer curve surrounds the inner curve. Let us denote

4.20

the velocity of the origin

4.21

If the initial curves have antipodal reflection symmetry then U(0,t)=0, because both integrals vanish. This means that 0 is a stagnation point, i.e. a fixed point of the Lagrangian path, for all time. An example of such a configuration is formed with two concentric ellipses (not necessarily aligned). The centre of vorticity coincides with the origin in these cases. The system in which z₂ has antipodal reflection symmetry is therefore

4.22

with ζ given by (4.15).

5. Inner ellipse

The system (4.22) has the remarkable property that, if the initial curve Γ₁ is an ellipse

then it remains an ellipse

5.1

where w_j(t) solve the ordinary differential equation (ODE) system

5.2

The proof of this fact is based on the following lemma.

Lemma 5.1 —

Let with |ζ₁|>|ζ₋₁|. Then

5.3

Proof. —

The proof of the lemma is based on a calculation done already in [9], but for the sake of completeness we present it below. The first observation is that

where

is the circular Hilbert transform. This follows from the properties of the logarithm and

which is obtained by integration by parts. Now we write

with

and expand

Raising δ(α,β) to a power k, we obtain

and the only non-zero contribution to the integral

comes from the second term, so

Therefore,

and (5.3) follows from

Now, this proof seems to work only if |ζ₁−1|+|ζ₋₂|<1, but the linear scaling property

valid for any , reduces the problem to this case. Indeed, if ζ₁=r e^iϕ and we choose c= (r+ϵ)⁻¹ e^−iϕ then

This concludes the proof of the lemma. ▪

Noting that

5.4

we can write (5.2) as

5.5

The conservation in time of A₁ (given in (5.4)) can be checked independently in (5.2). The system (4.22) now is reduced to the ODE system (5.5), where ζ is obtained from (4.15), coupling the ODE to equation (4.14). Kirchhoff ellipses are obtained by turning off the coupling, i.e. setting ω₂=0. The ODE system reduces further by considering the variable

5.6

This is a geometric quantity (see (3.2) and (3.3)):

The system (5.5) implies

In view of (5.4)

5.7

and therefore the equation for w is self-contained:

5.8

The variables w₁ and w₂ are easily obtained once w is known. In view of the geometric interpretation, we expect |w|=1 to be an invariant circle for the ODE. Indeed,

5.9

This shows that |w|=1 is an invariant circle for the equation. Moreover, in view of (5.4), if this set attracts a trajectory from inside (|w|<1), this means that the inner ellipse evolves in time and degenerates into a line . Indeed,

and |w|=1 implies b₁=0 and (because A₁=a₁b₁ is finite). This can happen only if

Let us write now

5.10

with and consider the evolution of w in a co-moving frame, i.e. we introduce the variable

5.11

Equation (5.8) becomes

5.12

and equation (5.9) becomes

5.13

We rescale time in order to have non-dimensional quantities. Setting τ=(ω₁/2)t we have

5.14

Writing u=x+iy, we arrive at

5.15

where we denoted

5.16

Recall that δ and Δ are computed from ζ, which is computed from the outer curve z₂. Our choice of variable u is motivated in the next section, where we compute an approximation of ζ explicitly, and obtain δ and Δ explicitly and, in addition, constant in time.

6. Two ellipses

We saw that, if the initial curve z₁(⋅,0) in (4.22) is an ellipse, then it remains an ellipse for all time, all be it with changing length of semi-axis. This is no longer the case for the evolution of z₂, unless the initial curve is a circle, in which case it stays a circle. If the initial data is an ellipse with small eccentricity, the evolution away from the ellipse will take a long time; the farther the curve and the smaller the eccentricity, the longer the time. More precisely, if we start with

6.1

the right-hand side of equation (4.14) (which is the same as the second equation of (4.22)) introduces higher harmonics. These are introduced not by the nonlinear term, but by the term , which is small. We therefore approximate the evolution of z₂ by projecting it on the elliptical modes. This is done in order to compute ζ(t) explicitly. Thus, if z₂=ζ₁ e^iα+ζ₂ e^−iα then we approximate

(Recall from (3.3) that circles correspond to ζ₂=0 with our orientation convention that |ζ₁|≥|ζ₂|.) With this, equation (4.14) with initial data (6.1) has solutions approximated by

6.2

with

6.3

Indeed, the approximate system is

and

so, from the second equation |ζ₂|²=|ζ₂(0)|², and substituting in the first equation we arrive at (6.3). Now, it is elementary to check that, if z₂=ζ₁ e^−iα+ζ₂ e^−iα, then ζ given by (4.15) is computed by

because the series converges if |ζ₂|<|ζ₁|. We have that

6.4

where

6.5

and, without loss of generality, we assumed that γ is real. Indeed, in view of (3.2) and (3.3),

where a₂ and b₂ are the major and minor semi-axes of Γ₂(0) and is the angle Γ₂(0) makes with the coordinate system. So, assuming that γ is real amounts to choosing the axis so that Ox is in the direction of the major semi-axis of Γ₂(0). If γ is real, then

6.6

and (6.4) follows from (6.3). Thus γ=(a₂−b₂)/(a₂+b₂) is time independent. The variable u defined in (5.11) describes the parameters of the inner ellipse Γ₁(t) in a frame which rotates with angular velocity −(1−γ²)(ω₁A₁+ω₂A₂)/2A₂. In particular,

6.7

gives the ratio a₁/b₁ of the major to minor semi-axes of Γ₁(t). The quantities δ and Δ defined in (5.16) and giving the coefficients in the system (5.15), which describes the evolution of u=x+iy, are

6.8

and

i.e.

6.9

They are constant because the lengths of the semi-axes of Γ₂(t), a₂ and b₂, are constant in time.

7. The ODE system

We investigate the system (5.15),

7.1

Recall that u=x+iy, where u parametrizes the inner ellipse via (5.1), (5.6) and (5.11). In view of (6.7), we are interested in x²+y²≤1. The quantities δ and Δ are constants, fixed by the outer ellipse via (6.8) and (6.9). The fixed points of (7.1) are given by

7.2

and

7.3

Proposition 7.1 —

The invariant set

for (7.1) attracts trajectories of (7.1) if and only if

7.4

By ‘attracts trajectories’, we mean that there exist (x₀,y₀) with such that the solution (x(t),y(t)) of (7.1) with initial data (x₀,y₀), satisfies

Proof. —

The quantity

7.5

is a conserved quantity for (7.1), as is easily verified. If we assume

then, from the hypothesis , , we derive a contradiction. Indeed, on the trajectory (x(t),y(t)), K takes the finite value K₀ computed at (x₀,y₀). Multiplying by (1−x²−y²) we obtain

and, on a sequence on which x(t_j)²+y(t_j)²→1 we deduce that ∈[−1,1] which is absurd. On the other hand, if (7.4) is satisfied, then the fixed points

7.6

lie on x²+y²=1. Analysing the linear stability we find that the linearized system is

7.7

and the fixed point (7.6) is stable if δy<0, and unstable if δy>0. By ODE theory, the stable fixed point has a non-empty open basin of attraction which therefore intersects x²+y²<1. This finishes the proof of the theorem except for the borderline case of |Δ/2δ|=1. This case necessitates a further study of the phase portrait of (7.1), which we also perform for other reasons. Before we do so, let us note that (7.4) holds if and only if

7.8

Note that (7.8) does not involve the aspect ratio of the inner ellipse. We investigate the system further. We take δ>0: in view of (6.8) and (7.8), this is the only possible case for instability if ω₂ has the same sign as ω₁. We need to find out how many solutions of the cubic equation in (7.2) lie in x²≤1. We look therefore for intersections of the curves f(x)=x−x³ and g(x)=δx²−Δx+δ in −1≤x≤1. The minimum of g is attained at x=Δ/2δ and is positive if g_min=δ−Δ²/4δ>0. The maximum of f is obtained at and equals . All intersections will be in 0≤x≤1. There will be two intersections if and only if the point is situated above the graph of the parabola y=g(x). The reason for this is that f(0)=0<g(0)=δ and f(1)=0<g(1)=2δ−Δ. In this case there will be two roots, . If the point is situated below the graph of the parabola, there will be no intersections, and if it is on the parabola, there will be one intersection point. The conditions are thus

7.9

When there are two solutions, then the smaller one x₁ is stable, and the larger one x₂ is unstable. Numerically, it is easy then to see that there is a homoclinic orbit connecting x₂ to itself and surrounding x₁. The circle x²+y²=1 is composed of two heteroclinic orbits going from the unstable fixed point on the circle to the stable one. There are also heteroclinic orbits connecting the unstable fixed point on the circle to x₂ and x₂ to the stable fixed point on the circle. If there is only one fixed solution then the previous picture simplifies, and, in addition to the two heteroclinic orbits on the unit circle, there are only heteroclinic orbits connecting the unstable fixed point on the circle to x₁=x₂, and connecting the latter to the stable fixed point on the circle. If there is no solution inside, then all orbits connect the unstable fixed point on the circle to the stable one. If |Δ/2δ|=1 then there are no orbits connecting the circle with the interior of the disc.

In view of the fact that |u|=|w| (see (5.11)), the upshot is that in all the cases obeying (7.4), when the unit circle attracts trajectories, it follows that where w is related to (5.1) by (5.6). Consequently, there is unbounded growth of the inner ellipse. Indeed, from the conservation of A₁ and from (5.7) it follows that , and that means, in view of (3.3), that the sum of lengths of semi-axes of the inner ellipse diverges. ▪

Competing interests

I declare I have no competing interests.

Funding

Research partially supported by NSF-DMS grants nos 1209394 and 1265132.