5.2 Lagrangian Mean Flow and Pseudomomentum
5.2.3 Impulse Budget of the GLM Equations
The impulse (also called Kelvin’s impulse or hydrodynamical impulse) is a classical concept in incompressible constant-density fluid dynamics going back to Kelvin [e.g. 30, 6]. In essence, the impulse complements the standard momentum budget whilst being based strictly on the vorticity of the flow. This can be a very powerful tool. We start by describing the classical impulse concept and then we go on to define a useful impulse for the GLM equations.
The classical impulse is a vector-valued linear functional of the vorticity defined by
impulse = 1
n−1
x×(∇×u)d V, (5.22)
where n >1 is the number of spatial dimensions, d V is the area or volume element, and the integral is extended over the flow domain. We are most interested in the two-dimensional case, in which
two-dimensional impulse =
(y,−x)∇×u d xd y. (5.23)
The impulse has a number of remarkable properties for incompressible perfect fluid flow. To begin with, the impulse is clearly well defined whenever the vorticity is compact, i.e. whenever the vorticity has compact support such that∇×u = 0 outside some finite region. If n=3 then this fixes the impulse uniquely, but if n=2 then the value of the impulse depends on the location of the coordinate origin unless the net integral of∇×u, which is the total circulation around the fluid domain, is zero. For example, in two dimensions the impulse of a single point vortex with circulationis equal to(Y,−X)where(X,Y)is the position of the vortex. This illustrates the dependence on the coordinate origin. On the other hand, two point vortices with equal and opposite circulations±separated by a distance d yield a coordinate-independent impulse vector with magnituded and direction parallel to the propagation direction of the vortex couple. To fix this image in your mind you can consider the impulse of the trailing vortices behind a tea (or coffee) spoon: the impulse is always parallel to the direction of the spoon motion.
The easily evaluated impulse integral in an unbounded domain contrasts with the momentum integral, which in the same situation is not absolutely convergent and therefore is not well defined [30, 40, 12]. For instance, in the case of the two- dimensional vortex couple the velocity field decays as 1/r2with distance r from the couple, which is not fast enough to make the momentum integral absolutely convergent. Thus a vortex couple in an unbounded domain has a unique impulse, but no unique momentum.
As far as dynamics is concerned, it can be shown that the unforced incompress- ible Euler equations in an unbounded domain conserve the impulse. The proof involves time-differentiating (5.22) and using integration by parts together with an estimate of the decay rate of u in the case of a compact vorticity field. Moreover, if the flow is forced by a body force F with compact support, then the time rate of change of the impulse is equal to the net integral of F. This follows from the vortic- ity equation in conjunction with a useful integration-by-parts identity for arbitrary vector fields with compact support:
F d V = −
x∇ ãF d V = 1 n−1
x×(∇×F)d V. (5.24) The integrals are extended over the support of F and the second term is included for completeness; it illustrates that∇ ãF and∇ìF are not independent for compact vector fields. Note that (5.24) does not apply to the velocity u because u does not have compact support. Now, in the tea spoon example the impulse of the trailing
vortex couple can be equated to the net force exerted by the spoon.4This illustrates how impulse concepts are useful for fluid–body interaction problems. For example, similar impulse concepts have been used to study the bio-locomotion of fish [19]
and of water-walking insects [12].
In a bounded domain the situation is somewhat different. Now the momentum integral for incompressible flow is convergent and in fact the net momentum is exactly zero because the centre of mass of an enclosed body of homogeneous fluid cannot move. The impulse, on the other hand, is nonzero and usually not constant in time anymore. This is obvious by considering the example of a vortex couple prop- agating towards a wall, which increases the separation d of the vortices and thereby increases5impulse. However, the instantaneous rate of change of the impulse due to a compact body force F is still given by the net integral of F. This works best if F is large but applies only for a short time interval, because then the boundary- related impulse changes are negligible during this short interval. Indeed, this kind of “impulsively forced” scenario gave the impulse its name. Finally, intermediate cases such as a zonal channel geometry are also possible, in which the flow domain is periodic or unbounded in x, but is bounded by two parallel straight walls in y. In this case the x-component of impulse is still exactly conserved under unforced flow, but not the y-component.
So now the question is whether the impulse concept can be applied to wave–
vortex interactions. The idea is to define a suitable mean flow impulse that evolves in a useful way under such interactions. This raises two issues. First, the classi- cal impulse concept is restricted to incompressible flow, i.e., if compressible flow effects are allowed, then most of the useful conservation properties of the impulse are lost. Still, the vortical mean flow dynamics, especially in the geophysically rel- evant regime of slow layer-wise two-dimensional flow, is often characterized by weak two-dimensional compressibility; a case in point is standard quasi-geostrophic dynamics in which the horizontal divergence is negligible at leading order. This suggests that two-dimensional impulse may still be useful. The second issue is the question as to which velocity field to use to form the impulse as in (5.23). For instance, one could base the GLM impulse on uL, but it turns out to be much more convenient to base the GLM impulse on uL −pinstead [17]. We therefore define the GLM impulse in the shallow water system as
I=
(y,−x)∇×(uL−p)d xd y=
(y,−x)qLh d xd y,˜ (5.25) where the integral extends over the flow domain, as before. Clearly,Iis well defined if qLhas compact support, which is a property that can be controlled from the initial
4More precisely, the time rate of change of the impulse equals the instantaneous force exerted by the spoon; time-integration then yields the final answer.
5It is a counter-intuitive fact that as d increases the impulse of the vortex couple increases even though its propagation velocity decreases! Indeed, the impulse is proportional to d and the velocity to 1/d.
conditions of the flow together with the mean material invariance of qL. Also,Iis obviously zero in the case of irrotational flow. This suggest that I is targeted on the vortical part of the flow, which is what we want, but the important question is how I evolves in time. The easiest way to find the time derivative ofI in the case of compact qL is by interpreting the integral in (5.25) as an integral over a material area that is strictly larger than the support of qL. The time derivative of such a material integral can then be evaluated by applying DLto the entire integrand, including d xd y. However, as both qL andhd xd y are mean material invariants the˜ only nonzero term comes from DL(y,−x)=(vL,−uL). After some integration by parts this yields
dI dt =
(uL−p)∇ ãuLd xd y+
(∇uL)ãpd xd y+remainder. (5.26)
Here the p contracts with uL and not with ∇, i.e. in index notation the second integrand is uLj,ipjwith free index i . Explicitly,
(∇uL)ãp=(uLxp1+vxLp2,uLyp1+vLyp2) (5.27) in terms of the pseudomomentum componentsp=(p1,p2).
The remainder in (5.26) consists of integrals over derivatives such as vLxvL = 0.5∂x(vL)2or(vLp2)x, which yield vanishing contributions in an unbounded domain if uL andpdecay fast enough with distance r . For example, a decay uL =O(1/r) or uL =O(1/r2)is sufficient, respectively, depending on whetherpis compact or not. We will assume thatpis compact in our examples (unless an explicit exception is made) and hence we can safely ignore this remainder. Likewise, the first term in (5.26) is due to compressibility and mean layer depth changes (via (5.19)), and we will assume that such compressible changes are relatively small, i.e. we assume that the second term in (5.26) is much bigger than the first. So, for practical purposes we approximate the impulse evolution by
dI dt = +
(∇uL)ãpd xd y. (5.28)
If the source term can be written as a time derivative of another quantity, then this would yield a conservation law. This is as far as we can go using the general exact GLM equations. Significantly more progress is possible if we turn to the ray tracing equations, which describe the evolution of a slowly varying wavetrain.