Instability Flow Angle

The following set of figures show spectra of ion density perturbations in the plane perpendicular to B₀ after computing the RMS over an appropriate time range. Each panel includes color-coded lines which aid in answering two questions fundamental to this chapter: 1) How does the flow angle of ion perturbations change with altitude?

2) Do thermal effects from the ITI significantly alter the flow angle beyond isothermal FBI? All five lines represent angles with respect to E₀× B0.

The first line, shown in magenta, gives the angle of relative drift velocity between electrons and ions, u_d = u_e− uⁱ. Theory predicts that the isothermal FBI growth rate should peak at the drift-velocity angle. In the absence of pressure gradients and inertia, assuming E0 = E0y and Bˆ 0 = B0z, the electron and ion drift componentsˆ are

u_ey =− eE₀ ν_em_e(1 + κ²_e) uex=−κ^euey = + κ_eeE₀

ν_em_e(1 + κ²_e) u_iy= + eE0

ν_im_i(1 + κ²_i) u_ix = +κ_iu_iy = + κ_ieE₀

ν_im_i(1 + κ²_i) The drift-velocity components are thus

u_dx= u_ex− u^ix

= +eE₀

 κe

m_eν_e(1 + κ²_e) − κi

m_iν_i(1 + κ²_i)



u_dy = u_ey− u^iy

=−eE⁰

 1

m_eν_e(1 + κ²_e)+ 1 m_iν_i(1 + κ²_i)



These components make an angle β = tan⁻¹(udy/udx) with the E0 × B⁰ (i.e., ˆx)

direction. Plugging in the above expressions yields

β = tan⁻¹



− (1 + κ²_i) + Θ²₀(1 + κ²_e) κe(1 + κ²_i)− Θ²0(1 + κ²_e)κi



(6.2)

where Θ₀ ≡ pmeν_e/m_iν_i as in Dimant and Oppenheim (2004). To be relevant to a simulation run, Θ₀ must use the simulated values of its parameters. The ion mass, m_i, is the physical ion mass but the electron mass, m_e, is inflated. Both ion and electron collision frequencies are as described in section 6.1. We also set ν_e to maintain the appropriate value of ψ for a given altitude, accounting for the artificial electron mass. At 107 km, β ≈ −9^◦; at 110 km, β ≈ −15^◦; at 113 km, β ≈ −24^◦. Plots of β made directly from u_dx:sim, u_dy:sim, and u_dz:sim in the sub-threshold run with E_y0 = 10 mV/m at each altitude (not shown) give these values directly.

The second line, shown in cyan, gives the predicted deflection of FBI+ITI per-turbations. Equation 34 of Oppenheim and Dimant (2004) is

tan 2χ_opt =−2κi(1 + ψ) 3− κ²i

Solving this equation for χ_opt and using the relation θ = χ + β yields an equation for θopt at a given altitude:

θ_opt = 1 2tan⁻¹



−2κi(1 + ψ) 3− κ²i

 + β

This angle represents the predicted angle of maximum growth of FBI+ITI perturba-tions. The values are θopt =−12^◦ at 107 km, θopt =−20^◦ at 110 km, and θopt =−32^◦ at 113 km. Note that, graphically, χ_opt is the difference between the magenta and cyan lines.

The third line, shown in white, actually represents three lines: the centroid of spectral power, with plus and minus one-σ uncertainty. The centroid of a 2-D distri-bution of points is a quantity familiar to most people. Calling it by its more colloquial

name, the center of mass, evokes an intuitive sense of the point at which the surface would balance on the head of a pin. Since spectral power is spread over a range of angles, the angular deflection of the centroid of spectral power represents flow angle between the wave vector, k, and E₀× B⁰.

if_ij is the total mass. These are just the components of the first moment of the distribution with respect to the radial co-ordinate r ≡ (x, y). The conversion from Cartesian to polar coordinates is simple:

hki = q

hxi²+hyi² and hθi = tan⁻¹(hyi / hxi).

In order to reduce the uncertainty in the centroid location, the analysis routine calculated the centroid for each image in the RMS time frame, calculated hθi as the mean centroid from that distribution, and calculated δhθi as the standard devia-tion of that distribudevia-tion. The standard deviadevia-tion is so small in all cases as to be imperceptible in the images.

It is worth noting that the centroid is a better measure of flow angle during growth than in saturation. During the growth stage, spectral amplitude is relatively isolated in both wavelength and angle, and the two peaks on either side of k_x = 0 are distinct. This means that the centroid of one of the peaks – the k_x > 0 peak in the following – represents the peak (k_x, k_y) value of linear growth. After saturation,

there is no longer a single peak wavelength that characterizes the instability. The centroid algorithm can still find the spectral center of mass but its value as a measure of flow angle is diminished. Nonetheless, it will serve as a visual guide.

Figure 6·22 shows RMS squared spectral amplitude in the plane perpendicular to B₀ during the growth stage and after saturation in 2-D runs. Each panel also shows the drift angle, β, the optimum FBI+ITI flow angle, θ_opt, and the flow-angle of the centroid, hθi.

The run at 107 km matches FBI+ITI theory well during growth: Despite the fact that χ_opt is only a few degrees at this altitude, hθi is within a few degrees of θ_opt. After saturation, the flow-angle magnitude increases by 1^◦ so that it sits clearly below both β and θ_opt. This increase is probably associated with the presence, then fading, of the ϕ≈ −45^◦ region in Figure 6·16.

At 110 km, hθi value sits approximately equidistant from β and θ^opt, indicating the possibility of some thermal effects but less than predicted. The magnitude of ϕ during growth in Figure 6·17 at angles near hθi is smaller, which suggests that the ITI simply does not enhance the FBI as much as in the run at 107 km. The transition from growth to saturation again carries an increase in flow-angle magnitude and the deviation from θ_opt is more extreme. Similarly to the run at 107 km, thermal effects appear to play a role in determining the saturated hθi value at 110 km. Unlike at 107 km, they increase the flow angle magnitude from less than θ_opt to greater than θ_opt.

At 113 km, hθi is approximately equal to β during growth but increases toward θ_opt in saturation. Physically, this implies that ion density perturbations at 113 km propagate at the angle from E₀× B⁰ predicted by isothermal FBI theory during the growth stage but become non-isothermal during the transition to saturation.

Figure 6·23 shows RMS spectral power in the plane perpendicular to B0 during

107 k m 110 k m 113 k m

Figure 6·22: RMS squared spectral amplitude in δnⁱ/n₀ during growth and after saturation in 2-D runs. Each panel spans 0 to +2π in k_x and −π to +π in k^y. Rows correspond to altitude, from 107 km (bottom) to 113 km (top). The left column shows the growth stage and the right column shows the saturation stage. In each panel, a ma-genta line indicates the drift angle, β, a cyan line indicates the optimal flow angle for the combined FBI+ITI, θ_opt, and white lines indicate the centroid angle,hθi, with ±σ uncertainties. The top of each panel lists the centroid angle. The color scale is identical to Figures 6·10 through 6·15.

107 k m 110 k m 113 k m

Figure 6·23: RMS squared spectral amplitude in δni/n₀ during growth and after saturation in 3-D runs. The figure layout is iden-tical to that of Figure 6·22.