Simulation Methods and Limitations

This chapter follows radar convention by defining flow angle as the angle between zeroth-order electron drift and LOS. Since electrons drift predominantly in the E₀× B₀ direction and radars can only observe Doppler shift from echoes propagating parallel or anti-parallel to their LOS, the flow angle is equivalently the angle between E₀ × B⁰ and the direction of wave propagation. Where the sign of flow angle is unspecified, the reader may assume that it is negative in a couner-clockwise sense – in terms of physical quantities, it points in a direction between E0× B⁰ and −E⁰.

Chapters 4 and 5 described the interaction of meter-scale waves with what many in the aeronomy community would call meso-scale and large-scale waves. This chap-ter focuses only on the dynamics of mechap-ter-scale waves driven by a constant electric field in a small patch of plasma. Whereas Chapters 4 and 5 assumed that thermal effects did not play an appreciable role in meter-scale irregularity development, this chapter allows thermal effects to alter the dynamics of meter-scale waves. To carry out this small-scale, non-isothermal analysis, the simulations described in this chap-ter used the pure-PIC version of EPPIC in both 2D (perpendicular to B₀) and 3D.

See Oppenheim and Dimant (2004) for a description of the advantages and

disad-136

the equatorial and high-latitude conditions with the same typical altitudinal profile of the neutral atmosphere taken from the MSIS-E-90 model available on Web (http://nssdc.gsfc.nasa.gov/space/model/models/

msis.html). The collision frequencies were calculated by using model formulas from Schunk and Nagy (2000).

Note that the neutral density and composition have latitudinal, seasonal, etc. variabilities roughly within 20%. Furthermore, some radar observations (Davies et al., 1997; St.-Maurice et al., 1999) suggest that the model results may to some extent overestimate the real neutral density. Thus the curves shown in Fig. 2 should serve just for orientation purposes and they do not describe actual ionospheric conditions at any given time and location.

Note that from Eq. (6b) we see that the wave phase velocity is always smaller than the ~ E

0

! ~ B

0

drift velocity.

This confirms the above assumption that electrons travel ahead of the wave (see Fig. 1), although at high altitudes where c " 1 and y is small the difference between ~ V

_Ph

and ~ V

₀

may be small.

3.3. Question 2: what drives the instabilities?

Here we address the above Question 2: what may cause the wave amplitude to grow, i.e., drive the instability? We will discuss physical driving mechanisms firstly for the Farley–Buneman instability and then for thermal instabilities.

3.3.1. What drives the FB instability?

In the long-wavelength limit, kV

0

" n

ⁱⁿ

, to first-order accuracy with respect to the small parameter kV

0

=n

in

, Eqs. (6) are common for all E-region instabilities. To

second-order accuracy, we should take into account wave pressure gradients, rdP ¼ rdðnT

^e

þ nT

ⁱ

Þ, and the ion inertia. They result in a slow temporal evolution of the considered quasi-stationary wave. Exponential growth of the density perturbations with time means linear instability. For isothermal or adiabatic plasma with no ion inertia (or large-scale gradients of the undisturbed background plasma density), the wave pressure gradients, rdP ¼ ðg

e

T

e

þ g

i

T

i

Þrdn, where g

_e;i

¼ 1 for isothermal particles and g

e;i

¼

⁵₃

for adiabatic particles, via ambipolar diffusion lead to damping of the initial density perturbations, dn (i.e. intrinsically there is no instability).

A small ion inertia, however, can drastically change the situation. In the wave frame, the ion inertia manifests itself via the convective term m

_i

ð ~ V

i

' rÞ ~ V

i

in the ion momentum Eq. (1b) and represents an additional kinetic ‘pressure’. Because ions speed up at the local density wells and slow down at the density hills (see Fig. 1) this additional ‘pressure’ is in anti-phase to the regular pressure and may reverse the sign of the total pressure gradients. This results in the Farley–Buneman (FB) instability.

3.3.2. What drives thermal instabilities?

The driving force for thermal instabilities is different.

It is the polarization wave electric field, d~ E, that plays a crucial role. Combined with the ambient electric field, E ~

0

, the wave electric field, d~ E, forms a wave-modulated total electric field which, via collisional friction, heats up electrons and ions. The average field ~ E

0

alone leads to the average frictional heating, while d~ E combined with E ~

0

lead to temperature modulations.

Fig. 3 shows two examples of the wavevector orientation. If the wavevector, ~ k, is pointing between the directions of ( ~ E

0

and ~ V

0

(bottom part of Fig. 3), modulations of the frictional heating (proportional to E ~

0

' d~ E) are in anti-phase to the density perturbations and may reverse the sign of the pressure perturbations, dP ’ ðT

ⁱ

þ T

^e

Þdn þ n

⁰

ðdT

ⁱ

þ dT

^e

Þ, driving the instabil-ity. If, however, ~ k is pointing between the directions ~ E

₀

and ~ V

₀

(see the top part of Fig. 3), modulations of the frictional heating are in phase to the density perturba-tions, so that the induced temperature modulations can only amplify the density gradients resulting in faster relaxation of initial density perturbations (i.e., increased damping).

Pressure reversal is the major driving mechanism of both the ET and IT instabilities. It is anti-symmetric with respect to the direction of the ~ E

0

! ~ B

0

drift. Linear thermal perturbations tend to zero as the wavevector ~ k approaches the directions parallel to either ~ E

0

(here d~ E ! 0) or ~ V

0

(here ~ E

0

' d~ E ! 0). The optimum flow angle for the thermal instabilities which maximizes dT / E ~

0

' d~ E lies in the bisector between the directions of ( ~ E

0

and ki ¼ Oⁱ=nin calculated for typical neutral density profiles and different geomagnetic field amplitudes: B0 ¼ 2:5 ! 10⁴nT for the equatorial E region (solid) and B0 ¼ 5 ! 10⁴nT for the high-latitude E region (dashed).

Y.S. Dimant, M.M. Oppenheim / Journal of Atmospheric and Solar-Terrestrial Physics 66 (2004) 1639–1654 1643

Figure 6·1: Figure 2 from Dimant and Oppenheim (2004), show-ing theoretical altitudinal profiles of ψ and κi at equatorial and high magnetic latitudes.

vantages of the pure-PIC version of the code. See Oppenheim et al. (2008) for a description of an improvement in parallelizing the 2-D version, and see Oppenheim and Dimant (2013) for a description of the 3-D version.

One major goal of the research presented in this chapter was to determine the change in FBI spectrum with altitude. Neutral density is a good proxy for altitude in the atmosphere, but EPPIC does not use neutral density as a simulation parameter, so the ion and electron collision frequencies, νi and νe, specify the equivalent altitude.

As Oppenheim and Dimant (2013) explain, the effective collision frequency during a simulation run differs, in general, from the input value. Selecting an input value that will produce an appropriate simulated value requires some care.

Figure 2 in Dimant and Oppenheim (2004), reproduced here in Figure 6·1, pro-vides a way to select collision frequencies corresponding to a desired altitude. The

first step is to identify an appropriate value of ψ⊥:sim for the desired altitude. Next, we identify the corresponding value of κ_i:sim = Ω_i/ν_i:sim from which we calculate ν_i:sim. We then use the definition ψ_⊥≡ ν^eν_i/Ω_eΩ_i, to calculate a value for ν_e:sim:

νe:sim= ψ⊥:sim

 Ω_iΩ_e:sim ν_i:sim



= ψ⊥:sim

 q_iq_eB₀² m_im_e:simν_i:sim



Often, ν_e:sim ≈ ν^i:sim, whereas ν_e ≈ 10νⁱ in the real E region. With these candidate values for ν_i:simand ν_e:simin hand, we run two types of simulations with sub-threshold electric fields to validate their values.

The process for validating ν_i:sim consists of running the simulator with a sub-threshold driving electric field, E_y0, and calculating the effective ion collision fre-quency from the ion Pedersen drift, uiP, via the zeroth-order drift relation νi:sim = q_iE_y0/m_iu_iP. The process for validating ν_e:sim consists of running the simulator with a small parallel electric field, Ek0, and calculating the effective electron col-lision frequency from the electron parallel drift, uek: νe:sim = |q^e|E^k0/meuek. The resulting collision frequencies are ν_i = 1022 s⁻¹ and ν_e = 965 s⁻¹, corresponding to 107 km; ν_i = 610 s⁻¹ and ν_e = 671 s⁻¹, corresponding to 110 km; ν_i = 369 s⁻¹ and νe = 491 s⁻¹, corresponding to 113 km. Many observations of FBI associated with the visible aurora – often called the “radar aurora” – assume that the echoes originate in a volume centered on 110 km. The effectively altitudes of the simula-tions presented here encompass that altitude to facilitate comparison to observasimula-tions.

Note that this chapter differs from Chapters 4 and 5, which used a constant value of ν_e for the electron fluid approximation in hybrid EPPIC. Table 6.1 lists the other parameters used in these simulations.

The ratio of ion mass to electron mass was artificially small for these simulation runs – a common practice in PIC simulations (cf. Chapter 3). Oppenheim and

Table 6.1: Simulation Parameters for Chapter 6

Symbol Value Unit Name

m_i 5.0× 10⁻²⁶ kg ion mass

m_i/m_e 1250 mass ratio

m_n 4.6× 10⁻²⁶ kg neutral mass

T_i 600 K initial ion temperature

T_e 1200 K initial electron temperature

T_n 300 K neutral temperature

n₀ 2× 10⁸ m⁻³ plasma density

ν_i 1022, 611, 369 s⁻¹ ion-neutral coll. freq.

ν_e 965, 671, 491 s⁻¹ electron-neutral coll. freq.

ψ⊥ 0.030, 0.013, 5.6× 10⁻³ anisotropy factor

h 107,110,113 km effective altitude

B_y0 5.0× 10⁻⁵ T magnetic field

E_z0 50.0 mV/m vertical electric field

L_x 40.96 m box length in X direction

dx 0.04, 0.08 m 2D, 3D cell size in X direction

L_y 40.96 m box length in Y direction

dy 0.04, 0.08 m 2D, 3D cell size in Y direction

Lt ≈ 460, ≈ 115 ms 2D, 3D time span

dt 1.75× 10⁻⁶, 3.0× 10⁻⁶ s 2D, 3D time step

Dimant (2004) noted that the simulation can use an artificially inflated electron mass as long as it maintains the electron and ion Hall and Pedersen drift rates, and the collision and thermalization rates. It must also keep the electron collision frequency large compared to the ion collision frequency, so that electron Landau damping does not become important.

In document Meter-scale waves in the E-region Ionosphere: cross-scale coupling and variation with altitude (Page 161-164)