Hodogram Analytics

Before addressing the problem of how to determine the downtail direction, we consider some of the general properties of hodograms and some of the properties specific to a propagating wave solution as discussed previously in this chapter. We begin by looking at the perpendicular components of a magnetic field oscillating in time t, an example of which was given previously by equations (4.1.17) and (4.1.18) but is restated here for convenience as

bx = b¯xcos(ωt),

by = b¯ycos(ωt−φ),

the phase shift betweenbx andby. The hodogram traced by the above time series will be elliptical (for non-zeroφ), with the eccentricity and inclination dependent on ¯bx, ¯by andφ. These properties can be determined by considering the ratio of the component time series in question, in our case by(t) to bx(t). This will define the gradient of the straight line from the origin to the point (bx(t), by(t)). The major/minor axes of the ellipse are defined by such a line, when the distance from the origin is maximised/minimised. To find these points consider the amplitude given by

q b2 x+b2y = q ¯ bx2cos2(ωt) + ¯by2cos2(ωt−φ) (4.3.1) Maximising (or minimising) this distance is equivalent to maximising its square. Hence considering d/dt = 0 yields d dt b 2 x+b2y

=−2ωb¯x2cos(ωt) sin(ωt)−2ωb¯y2cos(ωt−φ) sin(ωt−φ) = 0,

⇒b¯2

xsin(2ωt) + ¯b2ysin(2ωt−2φ) = 0,

⇒b¯2

xsin(2ωt) + ¯b2y(sin(2ωt) cos(2φ)−cos(2ωt) sin(2φ)) = 0,

⇒(sin(2ωt)) ¯b2 x+ ¯b2ycos(2φ) −(cos(2ωt)) ¯b2 ysin(2φ) = 0, ⇒tan(2ωt) = ¯ b2 ysin(2φ) ¯ b2 x+ ¯b2ycos(2φ) , ⇒t= 1 2ωtan −1      sin(2φ) ¯ b2 x ¯ b2 y + cos(2φ)      + nπ 2ω, (4.3.2)

forn= 0,1, wheretis the time at which the maximum or minimum is reached. Hence, the length of the major/minor ellipse axes is determined by substituting both solutions for the time in equation (4.3.2) into equation (4.3.1). Until this substitution it is unknown which root corresponds to the major/minor axes. No obvious simplification can be made in the resulting expression for the axes lengths in terms ofφ, and hence such an expression yields no further benefit than the separate equations (4.3.1) and (4.3.2). To find the gradient consider the ratio of by tobx given by

by bx = ¯ bycos(ωt−φ) ¯ bxcos(ωt) .

Then the time in equation (4.3.2) is again substituted into the above expression to obtain the gradient. As for the axes lengths, the resulting expression for the gradient in terms

ofφ is no more illuminating than the separate expressions and is thus omitted. With the above forms, the lengths and gradients of the major/minor ellipse axes are determined. We can also consider the relation between the perpendicular magnetic field hodogram and the hodograms of other components in the previously used tailward travelling wave model of Section 4.1.1. The following results only apply to the form of the wave assumed in that section, namely propagating in y but having an incident and reflected wave in the

x direction. In a similar manner to the above general expressions for the perpendicular magnetic field components, we can express the remaining components as

ux = u¯xsin(ωt), (4.3.3)

uy = u¯ysin(ωt−φ), (4.3.4)

bz = b¯zsin(ωt−φ), (4.3.5) with definitions consistent with those given above. We will now show that the perpendicu- lar velocity field components will trace a hodogram of the same inclination and eccentricity as the perpendicular magnetic field components, perhaps a surprising result at first glance. In this model where we have assumed a propagating wave solution, the amplitudes of these components are related (which would not necessarily be the case in a general system). Missing from the system of equations (4.1.22)-(4.1.27) is the definition of the amplitude of uy. It was disregarded before since it did not provide another independent quantity to aid with the solution of the system. However we state it here as

¯ uy =± ω B0kz ky kx q A2_i +A2 r−2AiArcos(2Φx) cos(kzz), to show that the amplitudes are related as

¯ uy ¯ ux = b¯_¯y bx ,

(see equations (4.1.22), (4.1.23) and (4.1.25)). Hence it can be shown that the expressions derived for the ellipse gradient for the perpendicular magnetic field components will be the same for the perpendicular velocity components, and hence their hodogram direction will coincide.

The relation to the perpendicular Poynting vector hodogram is more subtle because of the dependence of the components ofS⊥ onbz. It can be shown however, that the inclination of the hodogram ellipse for S⊥ will be the same as those of u⊥ and b⊥. Consider the

ratio of the components of S⊥ as Sy(t) Sx(t) = uy(t)bz(t) ux(t)bz(t) = uy(t) ux(t) .

This implies that a line drawn to a point on the u⊥ hodogram (and equally for b⊥) at time sayt0, will have the same gradient as the line drawn to a point on the S⊥ hodogram at t0. This however, does not prove that the hodograms are aligned, because the centre

of the S⊥ ellipse is offset from the origin. Figure 4.3.1 shows model hodograms of u⊥ and S⊥ to express this concept. The locations labelled A-H in Figure 4.3.1(a) map to the locations of the same letter in Figure 4.3.1(b), with the same slope to the origin as indicated by the red dashed lines. It can be seen that locations B and F, where ux = 0 correspond to Sx = 0, and locations D and H, where uy = 0, map toSy = 0. The latter locations also have bz = 0 (and hence Sx = Sy = 0) because uy and bz are in phase in this model. Indeed, it is this phase relation that gives Sy as purely positive such that positive yˆ is the downtail direction. Since the Poynting vector components have double the frequency of the velocity/magnetic field components, one orbit of u⊥ traces out two orbits ofS⊥. To show that the inclination of the hodograms is the same, we have to show that the gradient of the line through S0 (the center of the S⊥ hodogram) to the ellipse apex (i.e. the major axis) is the same as the major axis of theu⊥ ellipse. To do this, we use the same method as previously, whereby we attempt to find the time which maximises the distance

(a)u⊥ (b)S⊥

Figure 4.3.1: Model hodograms for (a) u⊥ and (b) S⊥ for the tailward travelling wave model. The lettered locations A-H map between the hodograms, showing the different locations on each for the same time.

q

(Sx−Sx0)2+ (Sy−Sy0)2, (4.3.6)

whereSx0 andSy0 are thex andy locations of theS⊥ ellipse center, i.e. S0 = (Sx0, Sy0).

Unfortunately it is not as simple algebraically to extract the time from this formulation, and hence to confirm the orientation of the S⊥ ellipse. However, it is simple to check numerically that in this system, the inclination of the S⊥ hodogram ellipse is the same as that of u⊥ and b⊥. In terms of the ellipticity, the S⊥ hodogram will differ from the others due to the fact that bz(t) enters the expression (4.3.6). This could be deter- mined as previously from the ratio ofSy toSx at the maximising/minimising times found numerically.

In this section, we have shown generally how to find the ellipticity and gradient of a hodogram for two harmonic time series. Considering the tailward travelling wave model presented in Section 4.1.1 we have then shown that in such a system, the hodograms for

b⊥,u⊥ and S⊥ will always be aligned (have the same ellipse gradient). The next section considers how, in general, we can determine the downtail direction using theS⊥hodogram.

4.3.2 Determining the Direction of Propagation Using the Poynting