A Novel Formulation of Point Vortex Dynamics on the Sphere: Geometrical and Numerical Aspects

Abstract

In this paper, we present a novel Lagrangian formulation of the equations of motion for point vortices on the unit 2-sphere. We show first that no linear Lagrangian formulation exists directly on the 2-sphere but that a Lagrangian may be constructed by pulling back the dynamics to the 3-sphere by means of the Hopf fibration. We then use the isomorphism of the 3-sphere with the Lie group SU(2) to derive a variational Lie group integrator for point vortices which is symplectic, second-order, and preserves the unit-length constraint. At the end of the paper, we compare our integrator with classical fourth-order Runge–Kutta, the second-order midpoint method, and a standard Lie group Munthe-Kaas method.

Appendices

Appendix A: Analysis of a Planar Vortex Integrator

In this appendix, we show that the integrator of Rowley and Marsden (2002) for point vortices in the plane shares a number of remarkable features with the Hopf integrator, which stems from the fact that both systems are derivable from a linear Lagrangian. Similar observations, but for the numerical integration of canonical Hamiltonian systems, were made by Brown (2006).

Decomposition into One-Step Methods

Rowley and Marsden (2002) start from the linear Lagrangian (1.3), which they discretize by setting

$$L_{\mathrm{d}}(z_0, z_1) = h L \biggl( (1-\alpha) z_0 + \alpha z_1, \frac{z_1 - z_0}{h} \biggr), $$

where α∈[0,1] is a real interpolation parameter. The equations of motion derived from this Lagrangian are given by

$$ \frac{z_{n+2} - z_n}{2h} = \alpha f(z_{n+\alpha}) + (1-\alpha) f(z_{n+1+\alpha}), $$

(A.1)

where z _n+α :=(1−α)z _n +αz _n+1 and f(z) is the right-hand side of the vortex equations (1.1). It turns out that for α=1/2, they can be written as the composition of a one-step method and its adjoint. To see this, we specialize to the case α=1/2 and use the fact that the original Lagrangian L is linear in the velocities to write

$$L_{\mathrm{d}}(z_0, z_1) = L( z_{1/2}, z_1) - L( z_{1/2}, z_0), $$

and we define L _d,+(z ₀,z ₁,h):=L(z _1/2,z ₁) and L _d,−(z ₀,z ₁,h):=−L(z _1/2,z ₀), so that L _d=L _d,++L _d,−. Consider the adjoint $L_{\mathrm{d}}^{*}$ of a discrete Lagrangian L _d, which is defined by $L_{\mathrm{d}}^{\ast}(z_{0}, z_{1}, h) := - L_{\mathrm{d}}(z_{1}, z_{0}, -h)$ (see Marsden and West 2001). Then, we have

$$L_{\mathrm{d}, +}^{\ast}(z_0, z_1, h) = L_{\mathrm{d}, -}(z_0, z_1, h), $$

and vice versa. This definition is motivated by the fact that the adjoint of the discrete Euler–Lagrange flow of a discrete Lagrangian is given by the discrete Euler–Lagrange flow of the adjoint discrete Lagrangian.

The composition of the discrete Euler–Lagrange flow of two discrete Lagrangians is given by the discrete Euler–Lagrange flow of a composition discrete Lagrangian that is the sum of the two original discrete Lagrangians. As a result, the discrete Euler–Lagrange flow for L _d is given by the composition of the discrete Euler–Lagrange flows for L _d,+ and its adjoint $L_{\mathrm{d},+}^{*}=L_{\mathrm{d}, -}$. These discrete flows can be viewed as one-step methods, and are typically only first-order accurate, but their composition is symmetric and therefore has even order of accuracy, and is, in particular, second-order accurate.

Lastly, we remark that for the point-vortex Lagrangian (1.3) the discrete Lagrangians L _d,+ and L _d,− coincide, which means that each of them is individually self-adjoint. As a result, the underlying one-step method is second-order. In fact, it can easily be seen that for α=1/2, the point-vortex equations (A.1) can be written as the composition of the implicit midpoint method

$$\frac{z_{n+1}-z_n}{h} = f(z_{n + 1/2}) $$

with itself. This method is clearly second-order accurate.

For the case of point vortices on the sphere the Lagrangians L _d,+ and L _d,− still coincide, but in order to recover the equations of motion (5.4) and to enforce the constraint 〈φ ⁿ⁺¹,φ ⁿ⁺¹〉=1, different constraint forces have to added to the discrete flow. As a result, the underlying one-step methods, which are the maps Φ _h and Ψ _h defined at the end of Sect. 5.2, no longer coincide and are individually only first-order accurate (unless the underlying Hamiltonian is $\mathbb{S}^{1}$-invariant), although their composition is second-order accurate.

The Choice α=0,1 for the Interpolation Parameter

The method (A.1) is implicit for all choices of α except α=0,1, in which case the equations become

$$ \frac{z_{n+2} - z_n}{2h} = f(z_{n+1}). $$

(A.2)

This method turns out to be the symmetric explicit midpoint method (see Hairer et al. 2002), which is well known to exhibit parasitic oscillatory solutions. These solutions can easily be observed in the dynamics of point vortices: in Fig. 10, we have plotted the energy error for a simulation of a four-vortex problem with vortex strengths Γ=(0.1,0.3,−0.2,−0.4) and initial conditions z=(0,0.5i,1,0.7+0.6i). For the simulation where α=0.9 the energy error is bounded, while for the simulation employing α=1.0 there is a clear linear drift in the energy error. The time step used for both simulations was h=0.1.

Fig. 10

For the four-vortex problem described in the text, the energy error exhibits a linear drift for the integrator with α=1 (solid line) but stays bounded whenever α≠1; here α=0.9 is shown (dashed line)

This is in clear contrast to the construction of variational integrators for nondegenerate Lagrangians, for which any choice of interpolation parameter α will result in a stable, second-order integrator.

Similar instabilities exist for the case of point vortices on the sphere: the discrete equations (5.6), for instance, exhibit the same instabilities as (A.2), despite being variational.

Appendix B: A Variational Integrator for Non-$\mathbb{S}^{1}$-invariant Hamiltonians

In Sect. 5.2 we were able to obtain the implicit midpoint version (5.10) of the Hopf integrator on $\mathbb{S}^{3}$ based on the assumption that the Hamiltonian H is invariant under the action of $(\mathbb{S}^{1})^{N}$ on $(\mathbb{S}^{3})^{N}$. When the Hamiltonian is not invariant, this simplification is no longer possible, and Eqs. (5.8) and (5.9) must be solved directly. In this appendix, we outline a strategy for doing so, based on the geometry of the group SU(2).

Implementing the Unit-Length Constraint: The Cayley Map

Given initial conditions (φ ⁿ⁻¹,φ ⁿ), we first compute the slack variables $d^{n}_{\alpha}$ using (5.9). We must now solve (5.8) for φ ⁿ⁺¹, and we need to impose the unit-length constraint (5.5). This can be done conveniently using the geometry of SU(2): we write the update map φ ⁿ↦φ ⁿ⁺¹ as

$$ \varphi^{n+1} = U^{n} \varphi^n, $$

(B.1)

where U ⁿ is an element of SU(2). This ensures that the length of φ ⁿ stays constant over time, since

$$\bigl(\varphi^{n+1}\bigr)^\dagger\varphi^{n+1} = \bigl(\varphi^n\bigr)^\dagger\bigl(U^{n} \bigr)^\dagger U^{n} \varphi^n = \bigl( \varphi^n\bigr)^\dagger\varphi^n, $$

so that, in particular, ∥φ ⁿ∥=1 implies that ∥φ ⁿ⁺¹∥=1.

Equations (5.8) for φ ⁿ⁺¹ can now be expressed as

$$ \operatorname{Re}\biggl[ \bigl(\varphi^n \bigr)^\dagger(\mathrm{i} \sigma_\alpha) \biggl(-\mathrm{i} \varGamma\bigl( U^{n} - I_{2\times2}\bigr){\varphi}^{n} + \frac{h}{2} D_{{\varphi}^\dagger}H\bigl(\varphi^{n+1/2}\bigr) \biggr) \biggr] = - d^n_\alpha, $$

(B.2)

where φ ^n+1/2 in the Hamiltonian can be expressed in terms of U ⁿ and φ ⁿ by

$$\varphi^{n+1/2} = \frac{1}{2} \bigl( \varphi^{n} + \varphi^{n+1} \bigr) = \frac{1}{2} \bigl( I + U^{n} \bigr) \varphi^n. $$

These equations can be solved for U ⁿ directly, but a computationally more advantageous approach is as follows. As long as the step size h is small, the update matrix U ⁿ will be in a neighborhood of the identity element in SU(2). We now parametrize that neighborhood by means of the Cayley transform $\mathrm{Cay}: \mathfrak{su}(2) \to \mathrm{SU}(2)$, given by

$$\begin{aligned} \mathrm{Cay}(A) & = (I + A) (I - A)^{-1}. \end{aligned}$$

That is, we search for an element $A^{n} \in\mathfrak{su}(2)$ such that U ⁿ=Cay(A ⁿ) will solve (B.2). The advantage is that $\mathfrak{su}(2)$ is a linear space, and that no constraints need to be imposed on A ⁿ, as the range of the Cayley map is automatically contained within SU(2). A standard non-linear solver can therefore be used to find A ⁿ. This is analogous to the approach used in Lee et al. (2009) to implement the unit-length constraint on S ² by updating the solution on the sphere using a SO(3) action that is parametrized by the Cayley transform from $\mathfrak{so}(3)$ to SO(3).

Computational Savings

Significant computational savings can be obtained by rewriting the Cayley map in a more convenient form. We recall from (2.6) that $\mathfrak{su}(2)$ is isomorphic with $\mathbb{R}^{3}$, and we denote the vector representation of A ⁿ by $\mathbf{a}^{n} \in \mathbb{R}^{3}$. A small calculation then shows that the Cayley transform can be expressed as

$$ U^{n} = \mathrm{Cay} \bigl(A^{n}\bigr) = \frac{1}{1 + \Vert \mathbf{a}^{n} \Vert^2} \bigl( \bigl(1 - \bigl\Vert \mathbf{a}^{n} \bigr\Vert^2\bigr) I + 2 A^{n} \bigr), $$

(B.3)

so that

$$U^{n} - I_{2\times2} = \frac{2}{1 + \Vert\mathbf{a}^{n} \Vert^2} \bigl( A^{n} - \bigl\Vert\mathbf{a}^{n} \bigr\Vert^2 I_{2\times2} \bigr). $$

The terms proportional to Γ in (B.2) can then be written as

$$\begin{aligned} &\operatorname{Re}\bigl[\varGamma\bigl(\varphi^n\bigr)^\dagger \sigma_\alpha\bigl( U^{n} - I_{2\times2}\bigr){ \varphi}^{n} \bigr] \\ &\quad =\frac{2\varGamma}{1 + \Vert \mathbf{a}^{n} \Vert^2} \operatorname{Re}\bigl[ \bigl( \varphi^n\bigr)^\dagger\sigma_\alpha\bigl( A^{n} - \bigl\Vert\mathbf{a}^{n} \bigr\Vert^2 I_{2\times2} \bigr){\varphi}^{n} \bigr] \\ &\quad = \frac{2\varGamma}{1 + \Vert \mathbf{a}^{n} \Vert^2} \bigl( \operatorname{Re}\bigl[\bigl(\varphi^n \bigr)^\dagger\sigma_\alpha A^{n} \varphi^n \bigr] - \bigl\Vert\mathbf{a}^{n} \bigr\Vert^2 \operatorname{Re}\bigl[\bigl( \varphi^n\bigr)^\dagger\sigma_\alpha \varphi^n \bigr] \bigr) \\ &\quad = \frac{- 2\varGamma}{1 + \Vert \mathbf{a}^{n} \Vert^2} \bigl( \mathbf{a}^{n} \times \mathbf{x}^{n} + \bigl\Vert\mathbf{a}^{n} \bigr\Vert^2 \mathbf{x}^{n} \bigr)_\alpha, \end{aligned}$$

(B.4)

where we have used the expression (2.11) for the Hopf fibration to write $x^{n}_{\alpha}= (\varphi^{n})^{\dagger}\sigma^{\alpha}\varphi^{n}$, as well as the identity

$$\begin{aligned} \bigl(\varphi^n\bigr)^\dagger\sigma_\alpha A^{n} \varphi^n & = \mathrm{i}\sum _{\beta=1}^3 \bigl(\mathbf{a}^{n} \bigr)_\beta\bigl(\varphi^n\bigr)^\dagger \sigma_\alpha\sigma_\beta\varphi^n \\ & = \mathrm{i} \bigl(\mathbf{a}^{n}\bigr)_\alpha- \bigl( \mathbf{a}^{n} \times\mathbf{x}^{n}\bigr)_\alpha, \end{aligned}$$

which follows easily from (2.7).

Similarly, the terms in (B.2) involving the derivatives of the Hamiltonian can be written using (4.13) as

$$\begin{aligned} &\operatorname{Re}\bigl( \bigl(\varphi^n\bigr)^\dagger(\mathrm{i} \sigma_\alpha) D_{{\varphi}^\dagger}H\bigl(\varphi^{n+1/2}\bigr) \bigr) \\ &\quad= \frac{1}{1 + \Vert \mathbf{a}_n \Vert^2} \bigl( \mathbf{x}^n \times\nabla {H}^{n+1/2}_{\mathbb{S}^2} - \bigl(\mathbf{a}^n \cdot \mathbf{x}^n\bigr) \nabla{H}^{n+1/2}_{\mathbb{S}^2} - \bigl( \mathbf{a}^n \times\mathbf{x}^n\bigr) \times \nabla{H}^{n+1/2}_{\mathbb{S}^2} \bigr)_\alpha, \end{aligned}$$

where $H_{\mathbb{S}^{2}}$ is the original point-vortex Hamiltonian (3.2).

Combining this expression with (B.4), we see that the first-order equations (B.2) for U ⁿ are equivalent to the following non-linear equation for a ⁿ:

$$\begin{aligned} &-2\varGamma\bigl( \mathbf{a}^{n} \times\mathbf{x}^{n} + \bigl\Vert\mathbf{a}^{n} \bigr\Vert^2 \mathbf{x}^{n} \bigr) + \frac{h}{2} \bigl( \mathbf{x}^{n} \times\nabla{H}^{n+1/2}_{\mathbb{S}^2} - \bigl( \mathbf{a}^{n} \cdot\mathbf{x}^{n}\bigr) \nabla {H}^{n+1/2}_{\mathbb{S}^2} \\ &\qquad{}- \bigl(\mathbf{a}^{n} \times\mathbf{x}^{n}\bigr) \times \nabla{H}^{n+1/2}_{\mathbb{S}^2} \bigr) \\ &\quad = -\bigl(1 + \bigl\Vert \mathbf{a}^n \bigr\Vert^2\bigr) \mathbf{d}^n. \end{aligned}$$

(B.5)

The first-order equations (5.9) can be rewritten in a similar fashion as a vector equation involving x ⁿ⁻¹,x ⁿ and a ⁿ⁻¹. However, as there is no need to solve these equations directly (they merely serve to determine the slack variable d ⁿ), we will not go in further detail.

Summary

To solve the discrete equations of motion (5.8) and (5.9) in the case of a non-$\mathbb{S}^{1}$-invariant Hamiltonian, we proceed as follows:

The advantage of computing φ ⁿ⁺¹ indirectly via a ⁿ is that (B.5) is a non-linear equation defined on $\mathfrak{su}(2)^{N}$. As this is a vector space, a standard non-linear solver can be used to solve (B.5). While Owren and Welfert (2000) developed an extension of Newton’s method that preserves the Lie group structure, it is much more computationally involved.