3.2 Cluster Modelling
3.2.1 Tailoring the Model
In order to model the Cluster observations we use specific input parameters given in
Clausen 08. Figure 10 from their paper displays a model of how the fundamental field
line frequency will vary with L-shell. Following the box model structure for the waveguide given in Figure 2.1.1, we take the inner boundary of the waveguide to be the plasmapause at ∼4 RE, and allow a radial extent (in x) of 10 RE to the magnetopause at∼ 14 RE.
the plasmapause to ∼ 5 mHz at the magnetopause. Assuming that the Alfv´en speed is proportional to the Alfv´en frequency, this allows the Alfv´en speed profile to be scaled to match this frequency change. We note that these frequencies are a little high for typical fundamental Alfv´en modes, probably due to lower plasma densities than normal which have elevated the normal Pc5 frequencies to the Pc4 band. Indeed, Clausen 08 suggest that it is due to the natural frequencies in this event matching the frequency of waves associated with back-streaming ions at the bow shock that the lowest frequency modes of the magnetosphere can be excited effectively by this method [Le and Russell, 1996]. The system is driven with the bz perturbation as described in Section 2.9, with a fre- quency of 17.2 mHz to match the dominant frequency in the Cluster data in Figure 3.1.1. Le and Russell [1996] developed a model to estimate the frequency generated by back- streaming ions at the bow shock, formulating the frequency in terms of the cone angle and the interplanetary magnetic field strength. Figure 8 from Clausen 08 shows that these parameters were relatively stable over the course of the event from 13:30 to 13:40 UT. Furthermore, even broadband frequency driving can give monochromatic signatures in a cavity or waveguide [Wright and Rickard, 1995a]. Both of these effects justify driving monochromatically. Our equilibrium model is chosen such that the driving frequency of 17.2 mHz is the second radial harmonic of the waveguide. Considering the radial funda- mental to be a quarter wavelength mode, with the prescribed boundary conditions the second harmonic has approximately three times the frequency of the fundamental. This implies a fundamental frequency of∼6 mHz.
In order to choose the wavenumber in the field-aligned direction kz, we again refer to the Alfv´en frequencies given in Figure 10 of Clausen 08. Through the Alfv´en dispersion relationωA=kzVA,kz can be adjusted withVAin order to match both the expected fun- damental frequency profile and the full dispersion relation of fast waveguide modes. This has been done in the previous Chapter in Section 2.7, where a fourth order Runge-Kutta method is used to solve the simplified system withky = 0. From this come the finalised values for the ky = 0 fast natural waveguide frequencies, the field-aligned wavenumber kz and the correctly scaled Alfv´en speed profile. (From this analysis we find dimensionless values of kz = 2.28, x0 = 1.514 and xc = 0.8. To get these in terms of RE, multiply
x0 and xc by 10, and divide kz by 10. The normalizing magnetic field and velocity were taken as 90 nT and 2350 km s−1 respectively, with an inner boundary Alfv´en speed of 2107 km s−1.) The theory of estimating the natural waveguide frequencies by looking at the ky = 0 modes was developed by Wright [1994] and tested in simulations by Rickard and Wright [1994].
The length in the z direction is now fixed by the choice of kz and the assumption that the modes have a fundamental structure inz. This is consistent with the observed phase shift of 180◦ in the azimuthal magnetic field component by between Cluster 3 and 4.
This infers that the satellites must straddle a node of by, one above and one below the magnetic equator. Results from a preliminary simulation using the above input parameters show that assuming a homogeneous medium in the z direction, at a satellite location modelling that of Cluster 3 (x = 0.05, y = 0.6 and z = 0.1 in dimensionless units), the by component has too small an amplitude in comparison to bz. The by component is small in the simulation due to the position of both satellites near to a node of by at the equator. Including an inhomogeneity in z i.e. a z dependent density profile, would create azdependent Alfv´en speed which could shift the turning point of the mode towards the equator. The low Alfv´en speed in this equatorial region tends to cause the mode’s phase structure to bunch up there, as seen in Figure 12 ofClausen 08showing the Alfv´en eigenfunctions. This is important as it would cause more rapid variations closer to the equator and could hence give a better match to the observed amplitudes.
To see the effects of including such an inhomogeneity, we consider resolving the eigenfunc- tions for the case whereky = 0 (hence∂/∂y = 0) andρ0 =ρ0(z). In Cartesian coordinates
the system is defined by equations (2.2.6), (2.2.3) and (2.2.5) without they dependence, given by µρ0(z) B0 ∂ux ∂t = ∂bx ∂z − ∂bz ∂x, (3.2.1) ∂bx ∂t = B0 ∂ux ∂z , (3.2.2) ∂bz ∂t = −B0 ∂ux ∂x . (3.2.3)
Considering a propagating solution inˆxof the formbz =bz(z)e−i(ωt−kxx)and rearranging yields ωµρ0(z) B0 ux = d dz(ibx) +kxbz, (3.2.4) − ω B0 (ibx) = dux dz , (3.2.5) ωbz = B0kxux. (3.2.6) Equation (3.2.6) can be used to eliminate bz from the (3.2.4). Normalising the magnetic field and velocity terms as discussed above, yields the system
d dz(ibx) = ω VA2(z)ux− k2 x ωux, (3.2.7) dux dz = −ω(ibx). (3.2.8)
where VA2(z) = B02/µρ0(z). We note that kx appears as kx2, so these equations apply equally well to a mode that has a standing structure inx. Also, since ρ0 =ρ0(z) there is
no preferred direction perpendicular toz, andxmay be taken as a general ‘perpendicular’ direction. To proceed, we transform the system into dipole coordinates, in order to real- istically express the variation of the Alfv´en speed along a field line. Figure 3.2.1 displays a typical field line in the northern hemisphere of a dipole system. We move from working inz to working with the latitudeλ. Derivatives are constructed using the chain rule as
d dz = dλ dz d dλ.
Figure 3.2.1: An example magnetic field line in the northern hemisphere of a dipole coordinate system, adapted from Wright [1987].
Restructuring the system in this way transforms equations (3.2.7) and (3.2.8) to
d dλ(ibx) = dz dλ ω V2 A(λ) −k 2 x ω ux, (3.2.9) dux dλ = −iωbx dz dλ. (3.2.10)
To specify the variation of the Alfv´en speed withλ, we consider the density profile provided
by Clausen 08, which once normalised gives
ρ0 =
r0
r
n
wherenis a density index set to 1 here,r is the geocentric distance to a point on the field line and r0 is the geocentric distance along the magnetic equator to the field line. The
normalisation by the Alfv´en speed at the equator yields a density of 1 atλ= 0, r=r0.
The magnetic field variation is taken to be a standard dipole variation of
B0(r, λ) = r0 r 3 1 + 3 sin2λ 1 2 ,
where the magnetic field strength has been normalised by the value at r =r0,λ = 0. r
can be eliminated from the equations usingr =r0cos2λto give
B0(r, λ) = 1 cos6λ 1 + 3 sin2λ 1 2 , ρ0 = 1 cos2λ.
Hence the normalised Alfv´en speed can be expressed as
VA2(λ) = B 2 0(λ) µρ0(λ) = 1 cos10λ 1 + 3 sin 2λ .
The final step is to evaluate the derivative dz/dλ. z here is essentially the path lengthS, which can be expressed as
S(λ) = r0 2√3
h
ppp2+ 1 + lnp+pp2+ 1i,
wherep=√3 sinλ, and hence we require dS/dλ, which can be easily evaluated.
To solve the system defined by (3.2.9) and (3.2.10), kx must be specified. The value is assumed based on a second radial harmonic as chosen for the modelling of this observation. With the boundary condition that ux has a node at the inner boundary and an antinode at the outer boundary, this givesλx= 4/3⇒kx= 4.712. When solving over the full field line length, the range of λ varies from the point where the field line reaches the Earth in the southern hemisphere, to the equivalent point in the northern hemisphere. This is calculated by considering where the radius is equal to 1. Assumingr0 = 5 consistent with
the L-shell of the Cluster satellites, with r = 1, using r =r0cos2λ gives the maximum
value of λto be 1.107, which implies λvaries over−1.107< λ <1.107.
antinodes. Using a fourth order Runge-Kutta method, the value of the frequency is in- cremented until these boundary conditions are met. In this scenario, thex direction can be thought of as a general perpendicular direction (we could have equally set ∂/∂x = 0 rather than ∂/∂y = 0), and hence we change subscripts from x to ⊥. Figure 3.2.2 plots
the fundamental eigenfunctions against distance along the field line, in a similar manner to Figure 12 from Clausen 08. The approximate locations of the Cluster satellites are denoted by the vertical dashed lines, and it is first of all evident that we see a 180◦ phase change in b⊥ (solid blue) between them, which is a feature of the observation. Secondly, the amplitude ofb⊥changes more quickly closer to the equator than for the homogeneous case (blue dashed line). Indeed at the satellite locations the amplitudes ofbz and b⊥ are comparable, whenVAvaries withzin accordance with the observations. The fundamental mode in z provides the desired phase shift, while including a density structure along the field produces the more rapidly changing amplitude profile close to the equator, which was the goal from the outset.
Including a z dependence in the density in the full time-dependent simulation increases the computing time, so we opt for a simpler approach of moving our simulation satellites inzto an equivalent phase and amplitude location, whilst retaining a density independent ofz for computational convenience. Figure 3.2.2 shows that it is justified to move further away from the equator along the field line closer to the amplitude peak of b⊥ in our simple model. Thinking in terms of nodal structure rather than height or distance from the equator, Figure 3.2.2 demonstrates that the Cluster satellites would lie close to the antinodes of b⊥. Hence moving closer to the antinodes in our original homogeneous z
structure, should simulate the amplitudes that would be found closer to the equator when there is inhomogeneity inz.