Stability Properties of Strongly Magnetized Spine Sheath Relativistic
  Jets
to appear with shortened appendicies in The Astrophysical Journal
Stability Properties of Strongly Magnetized Spine Sheath Relativistic Jets
Philip E. Hardee
Department of Physics & Astronomy, The University of Alabama, Tuscaloosa, AL 35487, USA
phardee@bama.ua.edu
ABSTRACT
The linearized relativistic magnetohydrodynamic (RMHD) equations describing a
uniform axially magnetized cylindrical relativistic jet spine embedded in a uniform axi-
ally magnetized relativistically moving sheath are derived. The displacement current is
retained in the equations so that effects associated with Alfvén wave propagation near
light speed can be studied. A dispersion relation for the normal modes is obtained. An-
alytical solutions for the normal modes in the low and high frequency limits are found
and a general stability condition is determined. A trans-Alfvénic and even a super-
Alfénic relativistic jet spine can be stable to velocity shear driven Kelvin-Helmholtz
modes. The resonance condition for maximum growth of the normal modes is obtained
in the kinetically and magnetically dominated regimes. Numerical solution of the dis-
persion relation verifies the analytical solutions and is used to study the regime of high
sound and Alfvén speeds.
Subject headings: galaxies: jets — gamma rays: bursts — ISM: jets and outflows —
methods: analytical — MHD — relativity — instabilities
1. Introduction
Relativistic jets are associated with active galactic nuclei and quasars (AGN), with black
hole binary star systems (microquasars), and are thought responsible for the gamma-ray bursts
(GRBs). In microquasar and AGN jets proper motions of intensity enhancements show mildly
superluminal for the microquasar jets ∼ 1.2 c (Mirabel & Rodriquez 1999), range from subluminal
(≪ c) to superluminal (. 6 c) along the M87 jet (Biretta et al. 1995, 1999), are up to ∼ 25 c along
the 3C 345 jet (Zensus et al. 1995; Steffen et al. 1995), and have inferred Lorentz factors γ > 100
in the GRBs (e.g., Piran 2005). The observed proper motions along microquasar and AGN jets
imply speeds from ∼ 0.9 c up to ∼ 0.999 c, and the speeds inferred for the GRBs are ∼ 0.99999 c.
Jets at the larger scales may be kinetically dominated and contain relatively weak magnetic
fields, e.g., equipartition between magnetic and gas pressure or less, but the possibility of much
stronger magnetic fields exists close to the acceleration and collimation region. Here general rel-
ativistic magnetohydrodynamic (GRMHD) simulations of jet formation (e.g., Koide et al. 2000;
Nishikawa et al. 2005; De Villiers, Hawley & Krolik 2003; De Villiers et al. 2005; Hawley & Krolik
2006; McKinney 2006; Mizuno et al. 2006) and earlier theoretical work (e.g., Lovelace 1976; Bland-
ford 1976; Blandford & Znajek 1977; Blandford & Payne 1982) invoke strong magnetic fields. In
addition to strong magnetic fields, GRMHD simulation studies of jet formation indicate that highly
collimated high speed jets driven by the magnetic fields threading the ergosphere may themselves
reside within a broader wind or sheath outflow driven by the magnetic fields anchored in the ac-
cretion disk (e.g., McKinney 2006; Hawley & Krolik 2006; Mizuno et al. 2006). This configuration
might additionally be surrounded by a less collimated accretion disk wind from the hot corona
(e.g., Nishikawa et al. 2005).
http://arxiv.org/abs/0704.1621v1
– 2 –
That relativistic jets may have jet-wind structure is indicated by recent observations of high
speed winds in several QSO’s with speeds, ∼ 0.1 − 0.4c, (Chartas, Brandt & Gallagher 2002,
Chartas et al. 2003; Pounds et al. 2003a; Pounds et al. 2003b; Reeves, O’Brien &Ward 2003). Other
observational evidence such as limb brightening has been interpreted as evidence for a slower external
sheath flow surrounding a faster jet spine, e.g., Mkn 501 (Giroletti et al. 2004), M 87 (Perlman et al.
2001), and a few other radio galaxy jets (e.g., Swain, Bridle & Baum 1998; Giovannini et al. 2001).
Additional circumstantial evidence such as the requirement for large Lorentz factors suggested by
the TeV BL Lacs when contrasted with much slower observed motions suggests the presence of
a spine-sheath morphology (Ghisellini, Tavecchio & Chiaberge 2005). At hundreds of kiloparsec
scales Siemignowska et al. (2007) have proposed a two component (spine-sheath) model to explain
the broad-band emission from the PKS 1127-145 jet. A spine-sheath jet structure has been proposed
based on theoretical arguments (e.g., Sol et al. 1989; Henri & Pelletier 1991; Laing 1996; Meier
2003). Similar type structure has been investigated in the context of GRB jets (e.g., Rossi, Lazzati
& Rees 2002; Lazzatti & Begelman 2005; Zhang, Wooseley & MacFadyen 2003; Zhang, Woosley &
Heger 2004; Morsony, Lazzati & Begelman 2006).
In order to study the effect of strong magnetic fields and the effect of a moving wind or
sheath around a jet or jet spine, I begin by adopting a simple system with no radial dependence of
quantities inside the jet spine and no radial dependence of quantities outside the jet in the sheath.
This “top hat” configuration with magnetic fields parallel to the flow can be described exactly
by the linearized relativistic magnetohydrodynamic (RMHD) equations. This system with no
magnetic and flow helicity is stable to current driven (CD) modes of instability (Istomin & Pariev
1994, 1996; Lyubarskii 1999). However, this system can be unstable to Kelvin-Helmholtz (KH)
modes of instability (Hardee 2004). This approach allows us to look at the potential KH modes
without complications arising from coexisting CD modes (see Baty, Keppens & Compte 2004) and
predictions can be verified by numerical simulations (Mizuno, Hardee & Nishikawa 2006).
This paper is organized as follows. In §2, I present the dispersion relation arising from a
normal mode analysis of the linearized RMHD equations. Analytical approximate solutions to
the dispersion relation for various limiting cases are given in §3. I verify the analytical solution
through numerical solution of the dispersion relation in §4. I summarize the stability results in
§5 and discuss the applicability of the present results in §6. Derivation of the linearized RMHD
equations is shown in Appendix A, derivation of the normal mode dispersion relation is presented
in Appendix B, and derivation of the analytical solutions is shown in Appendix C.
2. The RMHD Normal Mode Dispersion Relation
Let us analyze the stability of a spine-sheath system by modeling the jet spine as a cylinder
of radius R, having a uniform proper density, ρj, a uniform axial magnetic field, Bj = Bj,z, and a
uniform velocity, uj = uj,z. The external sheath is assumed to have a uniform proper density, ρe, a
uniform axial magnetic field, Be = Be,z, a uniform velocity ue = ue,z, and extends to infinity. The
sheath velocity corresponds to an outflow around the central spine if ue,z > 0 or represents backflow
when ue,z < 0. The jet spine is established in static total pressure balance with the external sheath
where the total static uniform pressure is P ∗e ≡ Pe + B2e/8π = P ∗j ≡ Pj + B2j /8π, and the initial
equilibrium satisfies the zeroth order equations. Formally, the assumption of an infinite sheath
means that a dispersion relation could be derived in the reference frame of the sheath with results
transformed to the source/observer reference frame. However, it is not much more difficult to derive
a dispersion relation in the source/observer frame in which analytical solutions to the dispersion
relation take on simple revealing forms. Additionally, this approach lends itself to modeling the
propagation and appearance of jet structures viewed in the source/observer frame, e.g., helical
structures in the 3C 120 jet (Hardee, Walker & Gómez 2005).
The general approach to analyzing the time dependent properties of this system is to linearize
– 3 –
the ideal RMHD and Maxwell equations, where the density, velocity, pressure and magnetic field
are written as ρ = ρ0 + ρ1, v = u + v1 (we use v0 ≡ u for notational reasons), P = P0 + P1, and
B = B0+B1, where subscript 1 refers to a perturbation to the equilibrium quantity with subscript
0. Additionally, the Lorentz factor γ2 = (γ0 + γ1)
2 ≃ γ20 + 2γ40u · v1/c2 where γ1 = γ30u · v1/c2.
The linearization is shown in Appendix A. In cylindrical geometry a random perturbation ρ1, v1
B1 and P1 can be considered to consist of Fourier components of the form
f1(r, φ, z, t) = f1(r) exp[i(kz ± nφ− ωt)] (1)
where flow is along the z-axis, and r is in the radial direction with the flow bounded by r = R. In
cylindrical geometry n is an integer azimuthal wavenumber, for n > 0 waves propagate at an angle
to the flow direction, and +n and −n give wave propagation in the clockwise and counter-clockwise
sense, respectively, when viewed in the flow direction. In equation (1) n = 0, 1, 2, 3, 4, etc.
correspond to pinching, helical, elliptical, triangular, rectangular, etc. normal mode distortions
of the jet, respectively. Propagation and growth or damping of the Fourier components can be
described by a dispersion relation of the form
n(βjR)
Jn(βjR)
n (βeR)
n (βeR)
. (2)
Derivation of this dispersion relation is given in Appendix B. In the dispersion relation Jn and H
are Bessel and Hankel functions, the primes denote derivatives of the Bessel and Hankel functions
with respect to their arguments. In equation (2)
χj ≡ γ2j γ2AjWj
̟2j − κ2jv2Aj
, (3a)
χe ≡ γ2eγ2AeWe
̟2e − κ2ev2Ae
, (3b)
β2j ≡
̟2j − κ2ja2j
̟2j − κ2jv2Aj
v2msj̟
j − κ2jv2Aja2j
 , (4a)
β2e ≡
̟2ex − κ2ea2e
̟2e − κ2ev2Ae
v2mse̟
e − κ2ev2Aea2e
. (4b)
In equations (3a & 3b) and equations (4a & 4b) ̟2j,e ≡ (ω − kuj,e)
and κ2j,e ≡
k − ωuj,e/c2
γj,e ≡ (1 − u2j,e/c2)−1/2 is the flow Lorentz factor, γAj,e ≡ (1 − v2Aj,e/c2)−1/2 is the Alfvén Lorentz
factor, W ≡ ρ+[Γ/ (Γ− 1)]P/c2 is the enthalpy, a is the sound speed, vA is the Alfvén wave speed,
and vms is a magnetosonic speed. The sound speed is defined by
where 4/3 ≤ Γ ≤ 5/3 is the adiabatic index. The Alfvén wave speed is defined by
1 + V 2A/c
where V 2A ≡ B20/(4πW0). A magnetosonic speed corresponding to the fast magnetosonic speed for
propagation perpendicular to the magnetic field (e.g., Vlahakis & Königl 2003) is defined by
vms ≡
a2 + v2A − a2v2A/c2
a2/γ2A + v
– 4 –
3. Analytical Solutions to the Dispersion Relation
In this section analytical solutions to the dispersion relation in the low frequency limit, in the
fluid and magnetic limits at resonance (maximum growth), and in the high frequency limit are
summarized. The analytical solutions are derived in Appendix C.
3.1. Low Frequency Limit
Analytically each normal mode n contains a single fundamental/surface wave (ω −→ 0, k −→ 0,
ω/k > 0) solution and multiple body wave (ω −→ 0, k > 0, ω/k −→ 0) solutions that satisfy the
dispersion relation. In the low frequency limit the fundamental pinch mode (n = 0) solution is
given by
uj ± vw
1± vwuj/c2
where the pinch fundamental mode wave speed
v2w ≈ a2j
v2msj
v2msj
, (6)
δ ≡ −
̟2e − κ2ev2Ae
v2msj
with |δ| ∝
∣k2R2
∣ << 1. In this limit δ is complex and this mode consists of a growing and damped
wave pair. The imaginary part of the solution is vanishingly small in the low frequency limit. The
above form indicates that growth, which arises from the complex value of δ, will be reduced as
(v2Aj/v
msj) −→ 1. The unstable growing solution is associated with the backwards moving (in the
jet fluid reference frame) wave.
In the low frequency limit the surface helical, elliptical, and higher order normal modes (n > 0)
have a solution given by
[ηuj + ue]± iη1/2
(uj − ue)2 − V 2As/γ2j γ2e
(1 + V 2Ae/γ
2) + η(1 + V 2Aj/γ
where η ≡ γ2jWj
γ2eWe and a “surface” Alfvén speed is defined by
V 2As ≡
γ2AjWj + γ
) B2j +B
4πWjWe
. (9)
In equation (9) note that the Alfvén Lorentz factor γ2Aj,e = 1 + V
Aj,e/c
2. Thus, the jet is stable to
n > 0 surface wave mode perturbations when
γ2j γ
e (uj − ue)
< γ2Ajγ
Ae +We/γ
) B2j +B
4πWjWe
. (10)
For example, with uj ≈ c >> ue, γ2e ≈ 1, γ2Aj >> γ2Ae ≈ 1, B2j >> B2e , and using γ2Aj =
1 +B2j /4πWjc
2 the jet is stable when
γ2j <
4πWec2
γ2Aj (11a)
– 5 –
or with Be = Bj , We = Wj , so that vA,j = vA,e, and with γA ≡ γA,e = γA,j the jet is stable when
γ2j γ
e (uj − ue)2 < 4γ2A(γ2A − 1)c2 . (11b)
Thus, the jet can remain stable to the surface wave modes even when the jet Lorentz factor exceeds
the Alfvén Lorentz factor.
In the low frequency limit the real part of the body wave solutions is given by
kR ≈ kminnmR ≡
v2msju
j − v2Aja2j
γ2j (u
j − a2j)(u2j − v2Aj)
× [(n+ 2m− 1/2)π/2 + (−1)mǫn] (12)
where n specifies the normal mode, m = 1, 2, 3, ... specify the first, second, third, etc. body wave
solutions, and
n(βjR)
n (βeR)
n (βeR)
In the absence of a significant external magnetic field and a significant external flow ǫn = 0 as
χe = γ
u2e − v2Ae
k2 = 0. In this low frequency limit the body wave solutions are either
purely real or damped, exist only when kminnmR has a positive real part, and with |ǫn| << 1 require
v2msju
j − v2Aja2j
γ2j (u
j − a2j )(u2j − v2Aj)
> 0 . (13)
Thus, the body modes can exist when the jet is supersonic and super-Alfvénic, i.e., u2j − a2j > 0
and u2j − v2Aj > 0, or in a limited velocity range given approximately by a2j > u2j > [γ2sj/(1+ γ2sj)]a2j
when v2Aj ≈ a2j , where γsj ≡ (1− a2j/c2)−1/2 is a sonic Lorentz factor.
3.2. Resonance
With the exception of the pinch fundamental mode which can have a relatively broad plateau
in the growth rate, all body modes, and all surface modes can have a distinct maximum in the
growth rate at some resonant frequency.
The resonance condition can be evaluated analytically in either the fluid limit where a >> VA
or in the magnetic limit where VA >> a. Note that in the magnetic limit, magnetic pressure
balance implies that Bj = Be. In these cases a necesary condition for resonance is that
uj − ue
1− ujue/c2
vwj + vwe
1 + vwjvwe/c2
, (14)
where vwj ≡ (aj , vAj) and vwe ≡ (ae, vAe) in the fluid or magnetic limits, respectively. When this
condition is satisfied it can be shown that the wave speed at resonance is
vw ≈ v∗w ≡
γj(γwevwe)uj + γe(γwjvwj)ue
γj(γwevwe) + γe(γwjvwj)
where γw ≡ (1 − v2w/c2)−1/2 is the sonic or Alfvénic Lorentz factor accompanying vwj ≡ (aj , vAj)
and vwe ≡ (ae, vAe) in the fluid or magnetic limits, respectively.
– 6 –
The resonant wave speed and maximum growth rate occur at a frequency given by
ωR/vwe ≈ ω∗nmR/vwe ≡
(2n + 1)π/4 +mπ
(1− ue/v∗w)
2 − (vwe/v∗w − uevwe/c2)
. (16)
In equation (16) n specifies the normal mode, m = 0 specifies the surface wave, and m ≥ 1 specifies
the body waves. In the limit of insignificant sheath flow, ue = 0, and using eq. (15) for v
w in eq.
(16) allows the resonant frequency to be written as
ω∗nmRj/vwe =
(2n + 1)π/4 +mπ
v2we/u
j + 2
vwevwj/γju
v2wj/γ
)]1/2
and this predicts a resonant frequency that is primarily a function of the sound and Alfvén wave
speeds in the sheath. The effect of sheath flow is best illustrated by assuming comparable conditions
in the spine and sheath, γwjvwj ∼ γwevwe, and assuming that γjuj >> γeue in which case
ω∗nmRj/vwe ∼
(2n + 1)π/4 +mπ
1− 2 (ue/uj) (1− v2we/c2)− (v2we − u2e) /u2j
The term ue/uj in the denominator indicates that the resonant frequency increases as the shear
speed, uj − ue, declines. In the limit
uj − ue
1− ujue/c2
vwj + vwe
1 + vwjvwe/c2
the resonant frequency ω∗nmR/vwe −→ ∞.
The resonant wavelength is given by λ ≈ λ∗nm ≡ 2πv∗w/ω∗nm and can be calculated from
λ∗nm ≡
(2n+ 1)π/4 +mπ
(v∗w − ue)
vwe − (vweue/c2)v∗w
R . (17)
Equations (15 - 17) provide the proper functional dependence of the resonant wave speed, frequency
and wavelength provided (ue/uj)
2 << 1 and (vwe/uj)
2 << 1.
With the exception of the n = 0, m = 0, fundamental pinch mode, a maximum spatial growth
rate, kmaxI , is approximated by
kmaxI R ≈ k∗IR ≡ −
ln |R| , (18)
where
|R| ≈
4 (ω∗nmR/vwe)
(1− 2ue/uj) + (ln |R| /2)2
(ln |R| /2)2
. (19)
Equations (18) and (19) show that the maximum growth rate is primarily a function of the jet
sound, Alfvén and flow speed through vwj/γjuj, and secondarily a function of the sheath sound,
Alfvén and flow speed through (ω∗nmR/vwe)
(1− 2ue/uj).
– 7 –
I can illustrate the dependencies of the maximum growth rate on sound, Alfvén and flow speeds
by using
ω∗nmR
(1− 2ue/uj) ≈
(1− 2ue/uj)
1− 2 (ue/uj) (1− v2we/c2)− (v2we − u2e)/u2j
] × [(2n + 1)π/4 +mπ]2
and if say ue = 0, then
|R|2 − 1
ln |R| ≈ 4
1− v2we/u2j
× [(2n + 1)π/4 +mπ] . (20)
Thus, |R| increases as ω∗nm increases for higher order modes with larger n and larger m and this
result indicates an increase in the growth rate for larger n and larger m. When the sound or Alfvén
wave speed, vwe, increases |R| increases. This result indicates an increase in the growth rate at
the higher resonant frequency accompanying an increase in the sound or Alfvén wave speed in the
sheath.
The behavior of the maximum growth rate as the shear speed, uj−ue, declines is best illustrated
by considering the effect of an increasing wind speed where (v2we − u2e)/u2j << 1 is ignored. In this
|R|2 − 1
ln |R| ≈ 4 [(2n + 1)π/4 +mπ] (21)
and |R| will remain relatively independent of ω∗nm even as ω∗nm −→ ∞ as the shear speed decreases.
This result indicates a relatively constant resonant growth rate as the shear speed decreases.
In the fluid limit decline in the shear speed ultimately results in a decrease in the growth rate
and increase in the spatial growth length. This decline in the growth rate is also indicated by
equation (8) which, in the fluid limit, becomes
ηuj + ue
1 + η
± i η
1 + η
(uj − ue) . (22)
Equation (22) applies to frequencies below the resonant frequency ω∗nm and directly reveals the
decline in growth rates as uj − ue −→ 0.
In the magnetic limit the resonant frequency ω∗nmR/vAe −→ ∞ as
uj − ue
1− ujue/c2
vAj + vAe
1 + vAjvAe/c2
. (23)
Here equation (8) indicates that the jet is stable when
γ2j γ
e (uj − ue)
< V 2As ,
and the jet will be stable as ω∗nm −→ ∞ when
γ2j γ
1− ujue/c2
< 2γ2Ajγ
v2Ae + v
(vAj + vAe)
1 + vAjvAe/c
, (24)
where I have used an equality in equation (23) in equation (8) to obtain equation (24). Equation
(24) indicates that a high jet speed relative to the Alfvén wave speed is necessary for instability.
For example, if vA ≡ vAj = vAe and ue = 0, the jet is stable at high frequencies provided
γ2j <
1 + v2A/c
γ4A . (25a)
– 8 –
This high frequency condition is slightly different from the low frequency stabilization condition
found when vA ≡ vAj = vAe and ue = 0 from equation (11b)
γ2j (uj/c)
2 < 4γ2A(γ
A − 1) . (25b)
Note that eqs. (25a & 25b) are identical in the large Lorentz factor limit. Equations (25) predict
that stabilization at high frequencies occurs at somewhat higher jet speeds than stabilization at
lower frequencies. Determination of stabilization at intermediate frequencies requires numerical
solution of the dispersion relation. A non-negligable postive external flow requires even higher jet
speeds for the jet to be unstable. Thus, a strongly magnetized relativistic trans-Alfvénic jet is
predicted to be KH stable and a super-Alfvénic jet can be KH stable.
3.3. High Frequency Limit
Provided the condition, eq. (14), for resonance is met, the real part of the solutions to the
dispersion relation in the high frequency limit for fundamental, surface, and body modes is given
uj ± vwj
1± vwjuj/c2
. (26)
and describes sound waves vwj = aj or Alfvén waves vwj = vAj propagating with and against the
jet flow inside the jet. Unstable growing solutions are associated with the backwards moving (in
the jet fluid reference frame) wave but the growth rate is vanishingly small in this limit.
4. Numerical Solution of the Dispersion Relation
The detailed behavior of solutions within an order of magnitude of the resonant frequency
and for comparable sound and Alfvén wave speeds must be investigated by numerical solution of
the dispersion relation. Analytical solutions found in the previous section can be used for initial
estimates and to provide the functional behavior of solutions. Numerical solution of the dispersion
relation also allows a determination of the accuracy and applicability of the analytical expressions
in §3.
In this section pinch fundamental, helical surface and elliptical surface, and the associated
first body modes are investigated in the fluid, magnetic and magnetosonic regimes. These modes
are chosen as they have been identified with structure seen in relativistic hydrodynamic (RHD)
numerical simulations or tentatively identified with structures in resolved AGN jets. For example,
trailing shocks in a numerical simulation (Agudo et al. 2002) and in the 3C 120 jet (Gómez et al.
2001) have been identified with the first pinch body mode. The development of large scale helical
twisting of jets has been attributed to or may be associated with growth of the helical surface mode,
e.g., 3C 449 (Hardee 1981) and Cygnus A (Hardee 1996) Additionally, the development of twisted
filamentary structures has been attributed to helical and elliptical surface and first body modes,
e.g., 3C 273 (Lobanov & Zensus 2001), M87 (Lobanov, Hardee & Eilek 2003), 3C 120 (Hardee,
Walker & Gómez 2005), and have been studied in RHD numerical simulations, e.g., Hardee &
Hughes (2003); Perucho et al. (2006).
4.1. Fluid Limit
In this section the basic behavior of the pinch (F) fundamental, helical (S) surface and elliptical
(S) surface modes is investigated: (1) as a function of varying sound speed in the external sheath
or jet spine for a fixed sound speed in the jet spine or external sheath and no sheath flow, (2) as a
function of equal sound speeds in the jet spine and external sheath for no sheath flow, and (3) as
– 9 –
a function of sheath flow for a relatively high sound speed equal in jet spine and external sheath.
In general only growing solutions are shown and complexities associated with multiple crossing
solutions are not shown. For all solutions shown the jet spine Lorentz factor and speed are set
to γ = 2.5 and uj = 0.9165 c. Sound speeds are input directly with the only constant being the
sheath number density. Total pressure and spine density are quantities computed for the specified
sound speeds. The adiabatic index is chosen to be Γ = 13/9 when 0.1 ≤ aj,e/c ≤ 0.5 consistent
with relativistically hot electrons and cold protons (Synge 1957). For sound speeds aj,e ∼ c/
3 the
adiabatic index is set to Γ = 4/3. Solutions shown assume zero magnetic field. Test calculations
with magnetic fields giving magnetic pressures a few percent of the gas pressure and Alfvén wave
speeds an order of magnitude less than the sound speeds give almost identical results.
In Figure 1 solutions in the left column are for a fixed jet spine sound speed aj = 0.3 c and in
the right column are for a fixed external sheath sound speed ae = 0.3 c. The solutions shown in
Figure 1 confirm the accuracy of the low frequency solutions to the pinch fundamental mode, eqs.
(5 & 6), and the helical and elliptical surface modes, eq. (8). Note that fast or slow wave speeds
are possible at low frequencies depending on whether η ≃ (γjae/γeaj)2 in eq. (8) is much greater
or much less than one, respectively. The numerical solutions to the dispersion relation show that
the maximum growth rate is primarily a function of the jet spine sound speed and only secondarily
a function of the external sheath sound speed as indicated by eqs. (18 - 20). Where a distinct
supersonic resonance exists, the resonant frequency is primarily a function of the external sheath
sound speed as predicted from eq. (16). The analytical expression for the resonant frequency for the
helical and elliptical surface modes provides the correct functional variation to within a constant
multiplier provided ae ≤ c/
3 and aj < c/3. A dramatic increase in the resonant frequency and
modest increase in the growth rate for larger jet spine sound speeds indicates the transition to
transonic behavior. Equation (15) for the resonant wave speed and equation (17) for the resonant
wavelength also provide a reasonable approximation to the functional variations provided ae ≤ c/
and aj < c/3. These results confirm the resonant solutions found in §3.2. At frequencies more than
an order of magnitude above resonance the growth rate is greatly reduced and solutions approach
the high frequency limiting form given by eq. (26). Note that eq. (26) allows only relatively high
wave speeds at high frequencies because aj ≤ c/
In Figure 2 the behavior of solutions to the fundamental/surface (left column) and associated
first body mode (right column) shows how solutions change as the sound speed increases in both the
jet spine and external sheath. Here I illustrate the transition from supersonic to transonic behavior
for no flow in the sheath. At low frequencies the modes behave as predicted by the analytic
solutions given in §3.1. The solutions show the expected shift to a higher resonant frequency that
is primarily a function of the increased external sheath sound speed and an accompanying increase
in the resonant growth rate that is primarily a function of the increased jet spine sound speed. The
resonance disappears as sound speeds approach c/
3 as the jet becomes transonic as predicted
by the resonance condition in §3.2. In the transonic regime high frequency fundamental/surface
mode growth rates and wave speeds are identical with wave speeds given by eq. (26). Provided the
jet is sufficiently supersonic, i.e., aj,e < 0.5 c, the maximum growth rate of the first body mode
is greater than that of the pinch fundamental mode, is comparable to that of the helical surface
mode, and is less than that of the elliptical surface mode. A narrow damping peak shown for the
helical first body (B1) solution when aj,e = 0.4 c is indicative of complexities in the body mode
solution structure. In the transonic regime growth of the first body mode is less than that of the
pinch fundamental, helical surface and elliptical surface modes.
Figure 3 illustrates the behavior of fundamental/surface and first body modes as a function
of the sheath speed for equal sound speeds in spine and sheath of aj,e = 0.4 c. For this value
of the sound speeds a sheath speed ue = 0 provides a supersonic solution structure baseline. At
low frequencies the surface modes behave as predicted by eq. (8), and the wave speed rises as ue
– 10 –
Fig. 1.— Solutions to the dispersion relation for pinch fundamental, helical surface and elliptical surface modes
for different sound speeds in the sheath (left column) and in the spine (right column) are shown for no sheath flow.
The real part of the wavenumber, krRj , is shown by the dashed lines and the imaginary part , kiRj , is shown by the
dash-dot lines as a function of the dimensionless angular frequency, ωRj/uj . For the pinch mode the vertical lines
indicate the maximum growth rate range. Otherwise, the vertical lines indicate the location of maximum growth.
Immediately under the dispersion relation solution panel is a panel that shows the relativistic wave speed, γwvw/c.
Line colors indicate the sound speed in units of c: (black) 0.10, (blue) 0.20, (cyan) 0.30, (green) 0.40, & (red) 0.577.
– 11 –
Fig. 2.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column)
and the first body (right column) modes are shown for equal sound speeds in spine and sheath and no sheath flow.
Real and imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 1. Locations
of the maximum growth rate are indicated by the vertical solid lines. A vertical arrow (helical B1) indicates a narrow
damping feature. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sound
speed in units of c: (black) 0.10, (blue) 0.20, (green) 0.40, & (red) 0.577.
– 12 –
Fig. 3.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column)
and first body (right column) modes as a function of the sheath speed for equal sound speeds in spine and sheath.
Real and imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 1. Locations
of the maximum growth rate are indicated by the vertical solid lines. Vertical arrows (helical B1) indicate damping
features. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sheath speed in
units of c: (black) 0.0, (blue) 0.20, (cyan) 0.35, (green) 0.40, & (red) 0.60.
– 13 –
increases. As ue increases the resonant frequency increases in accordance with eq. (16). On the
other hand, the growth rate at resonance does not vary significantly in accordance with eqs. (18 &
19). When the sheath speed exceeds the sound speed, solutions make a transition from supersonic
to transonic structure. Note that the transition point between supersonic and transonic behavior
is similar but not identical for the helical and elliptical surface modes, i.e., ocurs at a slightly
lower sheath speed for the elliptical mode. The first body modes also show an increase in resonant
frequency with little change in the maximum growth rate provided the sheath speed remains below
the sound speed. A significant damping feature in the helical first body (B1) panel, is found. While
a similar damping feature was not found for the pinch and elliptical first body mode, this does not
indicate a significant difference as the root finding technique does not find all structure associated
with the body modes. The body mode solution structure is complex with multiple solutions not
shown here and modest damping or growth can occur where solutions cross, e.g., Mizuno, Hardee
& Nishikawa (2006). When the sheath speed exceeds the sound speed the maximum body mode
growth rate delines significantly. This result is quite different from the transonic solution behavior
illustrated in Figure 2 when aj,e = 0.577 c for no sheath flow. Thus, sheath flow effects stability
of the relativistic jet beyond that accompanying an increase to the maximum sound speed in the
absence of sheath flow. The reduction in growth of the body modes in the presence of sheath flow
provides the relativistic jet equivalent of non-relativistic transonic/subsonic jet solution behavior.
At the higher frequencies wave speeds are identical, with wave speeds given by eq. (26). Note that
the high frequency wave speeds are nearly independent of ue.
4.2. Magnetic Limit
In this subsection the basic behavior of pinch, helical and elliptical modes is investigated: (1)
as a function of varying Alfvén speed in the external sheath or jet spine for a fixed Alfvén speed
in the jet spine or external sheath and no sheath flow, (2) as a function of equal Alfvén speeds
in the jet spine and external sheath for no sheath flow, and (3) as a function of sheath speed for
a relatively high Alfvén speed equal in jet spine and external sheath. In general only growing
solutions are shown and complexities associated with multiple crossing solutions are not shown.
For all solutions shown the jet spine Lorentz factor and speed are set to γ = 2.5 and uj = 0.9165 c.
Alfvén speeds are on the order of two magnitudes larger than the sound speed and are determined
by varying the sound speeds but with a gas pressure fraction on the order of 0.01% of the total
pressure. Only the sheath number density is held constant. The adiabatic index is set to Γ = 5/3
when aj,e/c << 0.1 consistent with low gas pressures and temperatures.
The solutions shown in Figure 4 confirm the theoretical predictions in the magnetic limit with
behavior depending on the Alfvén speed like the behavior found for the sound speed (see Figure
1). The pinch fundamental mode (not shown) has a growth rate almost entirely dependent on
sound speeds and is negligable in the magnetic limit as predicted by eq. (6). In Figure 4 solutions
in the left column are for a fixed jet spine Alfvén speed vAj = 0.3 c and in the right column are
for a fixed external sheath Alfvén speed vAe = 0.3 c. The solutions shown confirm the accuracy
of the low frequency solutions for helical and elliptical surface modes given by eq. (8). Note that
low frequency wave speeds can be high or low depending on the values of η = γ2jWj/γ
eWe, VAe/γe
and VAj/γj . The numerical solutions to the dispersion relation show that the maximum growth
rate is primarily a function of the jet spine Alfvén speed and only secondarily a function of the
external sheath Alfvén speed as predicted by eqs. (18 - 20). The resonant frequency is primarily a
function of the external sheath Alfvén speed as predicted by eq. (16). The analytical expression for
the resonant frequency of the helical and elliptical surface modes provides the correct functional
variation to within a constant multiplier provided vAj,e < 0.5 c. Decrease in the growth rate
for jet sheath Alfvén speeds vAe > 0.5 c indicates the transition towards trans-Alfvénic behavior.
Equation (15) for the resonant wave speed and equation (17) for the resonant wavelength also
– 14 –
Fig. 4.— Solutions to the dispersion relation for pinch fundamental, helical surface and elliptical surface modes
for different Alfvén speeds in the sheath (left column) and in the spine (right column) are shown for no sheath flow.
Sound speeds are aj,e ∼ 0.01vAj,e. As in Figures 1 - 3, the real part of the wavenumber, krRj , is shown by the
dashed lines and the imaginary part , kiRj , is shown by the dash-dot lines as a function of the dimensionless angular
frequency, ωRj/uj . The vertical lines indicate the location of maximum growth. The underlying panel shows the
relativistic wave speed, γwvw/c. Line colors indicate the Alfvén speed in units of c: (black) 0.10, (cyan) 0.30, (green)
0.50, & (red) 0.80.
provide a reasonable approximation to the functional variations for vAj,e < 0.5 c. At frequencies
more than an order of magnitude above resonance the growth rate is greatly reduced and solutions
approach the high frequency limiting form given by eq. (26). The surface modes have relatively
slow wave speeds, γwvw/c < 1 at high frequencies when the Alfvén wave speed vAj > 0.5 c. Unlike
the fluid case, the helical and elliptical surface modes are stabilized for Alfvén speeds somewhat in
excess of vAj,e ∼ 0.8 c in accordance with eqs. (8 & 24).
In Figure 5 the behavior of solutions to the pinch fundamental mode is shown in addition
to the helical and elliptical surface (left column) and associated first body modes (right column)
and the figure shows how solutions change as the Alfvén speed increases in both the jet spine and
external sheath. The sound speed is aj,e = 0.2 c for the pinch fundamental mode panel in order to
– 15 –
Fig. 5.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column)
and the first body (right column) modes are shown for equal sound speeds in jet and sheath and no sheath flow.
Pinch fundamental mode sound speed is aj,e = 0.2 c. Sound speeds for all other cases are aj,e ∼ 0.01vAj,e. Real and
imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 4. Locations of the
maximum growth rate are indicated by the vertical solid lines. A vertical arrow (elliptical B1) indicates a narrow
damping feature. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the Alfvén
speed in units of c: (black) 0.10, (blue) 0.20, (green) 0.40, & (red) 0.60.
– 16 –
Fig. 6.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column)
and the first body (right column) modes are shown for equal sound speeds in jet and sheath for different sheath
flow speeds. The pinch fundamental mode sound speed is aj,e = 0.2 c. Sound speeds for all other cases are
aj,e ∼ 0.01vAj,e = 0.005 c. Real and imaginary parts of the wavenumber as a function of angular frequency are
shown as in Figure 4. Locations of the maximum growth rate are indicated by the vertical solid lines. A vertical
arrow indicates low frequency damping of the pinch B1 solutions. The underlying panel shows the relativistic wave
speed, γwvw/c. Line colors indicate the sheath speed in units of c: (black) 0.0, (blue) 0.20, (cyan) 0.30, (green) 0.40,
& (red) 0.60.
– 17 –
illustrate the mode behavior with increasing Alfvén speed. Sound speeds for all body modes and
for helical and elliptical surface modes are aj,e ∼ 0.01vAj,e. Here the transition from super-Alfvénic
towards trans-Alfvénic behavior for no flow in the sheath is illustrated. At low frequencies the
modes behave as predicted by the analytic solutions given in §3.1. The growth rate of the pinch
fundamental mode is reduced as the Alfvén speed increases as predicted by eq. (6). The surface
and body mode solutions show the expected shift to a higher resonant frequency that is primarily
a function of the increased sheath Alfvén speed and an accompanying increase in the resonant
growth rate that is primarily a function of the increased spine Alfvén speed. The resonance moves
to higher frequency but the maximum growth rate is reduced for Alfvén speeds vAj,e > 0.60 c
and all modes become stable at higher Alfvén speeds in accordance with eqs. (8 & 24). At high
frequencies wave speeds are given by eq. (26). Provided the jet is sufficiently super-Alfvénic, i.e.,
vAj,e < 0.6 c, the maximum growth rate of the first body mode is much greater than that of the
pinch fundamental mode, is comparable to that of the helical surface mode, and is less than that
of the elliptical surface mode. A narrow damping peak shown for the elliptical body mode (B1)
solution when vAj,e = 0.6 c indicated by the arrow is indicative of complexities in the body mode
solution structure.
Figure 6 illustrates the behavior of fundamental/surface and first body modes as a function
of the sheath speed for an equal Alfvén speed in spine and sheath of vAj,e = 0.5 c. For this value
of the Alfvén speeds a sheath speed ue = 0 provides a super-Alfvénic solution structure baseline.
The sound speed is aj,e = 0.2 c for the pinch fundamental mode panel in order to illustrate the
mode behavior with increasing sheath speed. Sound speeds for all other cases are aj,e ∼ 0.01vAj,e.
Solutions in the first pinch body mode panel show the damping solution as opposed to the purely
real solution at the lower frequencies (indicated by the arrow). At higher frequencies the body mode
is growing. At low frequencies the surface modes behave as predicted by the analytic solutions given
in §3.1 and the growth rate of the surface modes decreases as ue increases. Additionally, the growth
rate at resonance decreases as expected for this relatively high Alfvén speed as the sheath speed
increases. At the higher frequencies wave speeds are identical, with wave speeds given by eq. (26).
Note that the high frequency wave speeds are relatively independent of ue. When the velocity shear
speed drops to less than the “surface” Alfvén speed, see eq. (11b), the helical and elliptical surface
modes and the first body modes are stabilized. This surface and body mode mode stabilization
occurs when sheath speeds exceed ue ∼ 0.5 c. However, note that the maximum pinch fundmental
mode growth rate is insensitive to the sheath speed and remains unstable at ue = 0.6 c even when
all other modes are stabilized.
4.3. A High Sound and Alfvén Speed Magnetosonic Case
In this subsection the basic behavior of the pinch fundamental, helical surface, elliptical surface
and associated first body modes is illustrated for different sheath speeds. The sheath speeds span a
solution structure from supersonic to transonic but still super-Alfvénic flow. Here the sound speed
in jet spine and external sheath are set equal with aj,e = 0.577 c and Alfvén speeds are set equal
with vAj,e = 0.5 c. The solutions for this case are shown in Figure 7. With no sheath flow the
fundamental/surface and first body modes show a typical supersonic and super-Alfvénic structure
albeit the pinch fundamental mode now has a maximum growth rate comparable to the helical
and elliptical surface modes as a consequence of the high sound speed. The associated first body
modes also have maximum growth rates comparable to the fundamental/surface modes. Increase
in the sheath speed results in a decrease in the growth rate of the helical and elliptical surface
modes at low frequencies as predicted by eq. (8). The low frequency growth rate of the pinch
fundamental also declines with increasing sheath speed. The resonant frequency increases with
increasing sheath speed as expected from the analytical and numerical studies performed in the
fluid and magnetic limits and the fundamental/surface modes take on a transonic structure for
– 18 –
Fig. 7.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column),
and the associated first body (right column) modes are shown for a maximal spine and sheath sound speed, aj,e =
0.577 c, and a slightly smaller spine and sheath Alfvén speed, vAj,e = 0.5 c, for different sheath flow speeds. As in
previous figures the real part of the wavenumber, krRj , is shown by the dashed lines, the imaginary part , kiRj ,
is shown by the dash-dot lines, and the vertical lines indicate the location of maximum growth. Arrows indicate
damping features. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sheath
speed in units of c: (black) 0.0, (blue) 0.20, (green) 0.40, & (red) 0.50. Fast and slow refer to the faster and slower
moving solutions and the yellow extension indicates a damped solution.
– 19 –
sheath speeds 0.4 c ≤ ue ≤ 0.1 c.
At high frequencies the fundamental/surface modes exhibit very high growth rates provided
sheath flow remains below the Alfvén speed. On the other hand, the maximum growth rate of the
first body modes declines as the sheath speed increases and is reduced severely when ue > 0.1 c.
This behavior is similar to what is found for non-relativistic jets as flow enters the transonic and
super-Alfvénic regime (Hardee & Rosen 1999). Additional increase in the sheath flow speed to
ue > 0.4 c results in a decrease in the growth rate of the fundamental/surface modes. Solutions
for the helical and elliptical surface modes shown in Figure 7 for a sheath speed ue = 0.5 c equal
to the Alfvén speed illustrate some of the complexity associated with barely super-Alfvénic flow.
Here limited growth is associated with both the slow and fast helical and elliptical surface solution
pair. At slower sheath speeds in the super-Alfvénic regime growth is associated with the slow
surface solution, i.e., backwards moving in the jet fluid reference frame. The yellow dash-dot
line extension at higher frequencies in the helical and elliptical surface panels indicates a damped
solution. Solutions were very difficult to follow in this parameter regime and it is possible that
some solutions were not found. When the sheath speed ue > 0.5 c all modes are stabilized.
A choice of Alfvén speeds greater than sound speeds results in a more magnetic like solution
structure like that shown in §4.2. A choice of Alfvén speeds more than a factor of two less than sound
speeds produces a more fluid like solution structure like that shown in §4.1. The more complicated
solution structure illustrated in Figure 7 only occurs for a relatively narrow range of high sound
speeds with similar or slightly lesser Alfvén speeds. In general, the detailed solution structure for
situations in which sound and Alfvén speeds are comparable must by examined individually, e.g.,
Mizuno, Hardee & Nishikawa (2006), and further investigation of these cases is beyond the scope
of the present paper.
5. Summary
The analytical and numerical work performed here provides for the first time a detailed analysis
of the KH stability properities of a RMHD jet spine-sheath configuration that allows for relativistic
motions of the sheath, sound speeds up to c/
3, and, by keeping the displacement current in the
analysis, Alfvén wave speeds approaching lightspeed and large Alfvén Lorentz factors. In the fluid
limit, the present results confirm an earlier more restricted low frequency analytical and numerical
simulation study performed by Hardee & Hughes (2003). Provided the jet spine is super-sonic and
super-Alfvénic internally and also relative to the sheath, the helical, elliptical and higher order
surface modes and the pinch, helical, elliptical and higher order first body modes have a maximum
growth rate at a resonant frequency. The pinch fundamental growth rate is significant only when
the sound speeds, aj,e ∼ c/
3. In general, the first body mode maximum growth rate is: greater
than the pinch fundamental mode, slightly greater than the helical surface mode, slightly less than
the elliptical surface mode, and occurs at a higher frequency than the maximum growth rate for
the fundamental/surface mode.
The basic KH stability behavior as a function of spine-sheath parameters is indicated by the
analytic low frequency surface mode solution and by the behavior of the resonant frequency. The
analytic surface mode solution valid at frequencies below resonance is given by
± iωi
[ηuj + ue]± iη1/2
(uj − ue)2 − V 2As/γ2j γ2e
(1 + V 2Ae/γ
2) + η(1 + V 2Aj/γ
where
V 2As ≡
γ2AjWj + γ
) B2j +B
4πWjWe
, (28)
– 20 –
and η ≡ γ2jWj
γ2eWe , V
A ≡ B2/4πW , W ≡ ρ + [Γ/ (Γ− 1)]P/c2 and γA ≡ (1 − v2A/c2)−1/2.
Equation (27) provides a temporal growth rate, ωi(k), and a wave speed, vw = ωr/k. The reciprocal
provides a spatial growth rate ki(ω), and growth length ℓ = k
i . Increase or decrease of the
growth rate, dependence on physical parameters and stabilization at frequencies/wavenumbers
below resonance is directly revealed by ωi in eq. (27). Note that higher jet Lorentz factors reduce
ωi through the dependence on η.
The resonant frequency is
(1− ue/v∗w)
2 − (vwe/v∗w − uevwe/c2)
, (29)
where v∗w is the wave speed at resonance, eq. (15). The resonant frequency increases as the sheath
sound or Alfvén wave speed, vwe ≡ (ae, vAe) increases and ω∗ −→ ∞ when the denominator
decreases to zero as
uj − ue
1− ujue/c2
vwj + vwe
1 + vwjvwe/c2
where vwj,e ≡ (aj,e, vAj,e) in the fluid and magnetic limits, respectively. Since eq. (27) applies below
resonance the overall behavior of the growth rate is indicated by ωi. Thus, growth rates decline to
zero as (uj − ue)2 − V 2As/γ2j γ2e −→ 0. The numerical analysis of the dispersion relation shows that
the pinch fundamental and all first body modes are comparably or more readily stabilized and thus
the jet is KH stable when
(uj − ue)2 − V 2As/γ2j γ2e < 0 . (30)
This stability condition takes on a particularly simple form when conditions in spine and sheath
are equal, i.e., Be = Bj, We = Wj, so that vA,j = vA,e, and with γA ≡ γA,e = γA,j
γ2j γ
e (uj − ue)
< 4γ2A
γ2A − 1
c2 (31)
indicates stability. This result implies that a trans-Alfvénic relativistic jet with γjuj & γAvA will
be KH stable, and that even a super-Alfvénic jet with γj >> γA can be KH stable.
6. Discussion
Formally, the present results and expressions apply only to magnetic fields parallel to an axial
spine-sheath flow in which conditions within the spine and within the sheath are independent of
radius and the sheath extends to infinity. A rapid decline in perturbation amplitudes in the sheath
as a function of radius, governed by the Hankel function in the dispersion relation, suggests that
the present results will apply to sheaths more than about three times the spine radius in thickness.
The relativistic jet is transonic in the absence of sheath flow only for spine and sheath sound
speeds ∼ c/
3. Only in this regime does the pinch fundamental have a significant growth rate and,
in general, we do not expect the pinch fundamental to grow significantly on relativistic jets. On
the other hand, the pinch first body mode can have a significant maximum growth rate and would
dominate any axisymmetric structure. The elliptical and higher order surface modes have increas-
ingly larger maximum growth rates at resonant frequencies higher than the helical surface mode,
and the maximum first body mode growth rates for helical and elliptical modes are comparable to
that of the surface modes. Nevertheless, we expect the helical surface mode to achieve the largest
amplitudes in the non-linear limit as a result of the reduced saturation amplitudes that accompany
the higher resonant frequency and shorter resonant wavelengths associated with the higher order
surface modes and all body modes.
– 21 –
In astrophysical jets we expect a toroidal magnetic field component, and possibly an ordered
helical structure and accompanying flow helicity. Jet rotation (e.g., Bodo et al. 1996), or a radial
velocity profile (e.g., Birkinshaw 1991) will modify the present results but will not stabilize the
helical mode. Two dimensional non-relativistic slab jet theoretical results, indicate that KH sta-
bilization occurs when the velocity shear projected on the wavevector is less than the projected
Alfvén speed (Hardee et al. 1992). In the work presented here magnetic and flow field are parallel
and project equally on the wavevector which for the helical (n = 1) and elliptical (n = 2) mode
lies at an angle θ = tan−1(n/kR) relative to the jet axis. Provided magnetic and flow helicity and
radial gradients in jet spine/sheath properties are not too large we expect the present results to
remain valid where uj,e and Bj,e refer to the poloidal velocity and field components.
KH driven normal mode structures move at less than the jet speed. The fundamental pinch
mode moves backwards in the jet frame at about the sound speed nearly independent of the sheath
properites and thus moves at nearly the jet speed in the source/observer frame. Low frequency and
long wavelength helical and higer order surface modes are advected with wave speed indicated by eq.
(27) and move slowly in the source/observer frame for light, i.e. η ≡ γ2jWj
γ2eWe < 1, and/or for
magnetically dominated flows. Higher frequency (above resonance) and shorter wavelength normal
mode structures move backwards in the jet frame at the sound/Alfvén wave speed, have a wave
speed nearly independent of the sheath properties, and can move slowly in the source/observer
frame only for magnetically dominated flows.
Where flow and magentic fields are parallel, current driven (CD) modes are stable (Isotomin
& Pariev 1994, 1996). Where magnetic and flow fields are helical CD modes can be unstable
(Lyubarskii 1999) in addition to the KH modes. CD and KH instability are expected to produce
helically twisted structure. However, the conditions for instability, the radial structure, the growth
rate and the pattern motions are different. For example, KH modes grow more rapidly when the
magnetic field is force-free (e.g., Appl 1996), and non-relativistic simulation work (e.g., Lery et
al. 2000; Baty & Keppens 2003; Nakamura & Meier 2004) indicates that CD driven structure is
internal to any spine-sheath interface and moves at the jet speed.
The differences between KH and CD instability can serve to identify the source of helical
structure on relativistic jets and allow determination of jet properties near to the central engine.
Perhaps the observation of relatively low proper motions in the TeV BL Lacs when intensity mod-
eling requires high flow Lorentz factors (Ghisellini et al. 2005) is an indication of a magnetically
dominated KH unstable spine-sheath configuration.
The author acknowledges partial support through National Space Science and Technology
Center (NSSTC/NASA) cooperative agreement NCC8-256 and by National Science Foundation
(NSF) award AST-0506666 to the University of Alabama.
A. Linearization of the RMHD Equations
In vector notation the relativistic MHD continuity equation, energy equation, and momentum
equation can be written as:
[γρ] +∇ · [γρv] = 0 , (A1)
γ2W − P
)− (v/c ·B)
γ2Wv +
v−(v ·B) B
= 0 , (A2)
v + v · ∇v
= −∇P − v
P + ρqE+
[j×B]
. (A3)
– 22 –
These equations along with Maxwell’s equations
∇ ·B = 0 ∇ ·E = 4πρq
∇×B = 1
E+ 4π
j ∇×E = −1
and assuming ideal MHD with comoving electric field equal to zero
E = −v ×B
provide the complete set of ideal RMHD equations. In the above W is the enthalpy, the Lorentz
factor γ = (1 − v · v/c2)−1/2, and ρ is the proper density. In what follows I will assume that the
effects of radiation can be ignored, the enthalpy is given by
W = ρ+
and the condition for isentropic flow is given by
+ v · ∇
= 0 .
The general approach to analyzing the time dependent properties of this system is to linearize
the ideal RMHD equations, where the density, velocity, pressure and magnetic field are written as
ρ = ρ0 + ρ1, v = u + v1 (we use v0 ≡ u for notational reasons), P = P0 + P1 E = E0 + E1, and
B = B0+B1, where subscript 1 refers to a perturbation to the equilibrium quantity with subscript
0. Additionally, W = W0 + W1, γ
2 = (γ0 + γ1)
2 ≃ γ20 + 2γ40u · v1/c2 and γ1 ≃ γ30u · v1/c2. It
is assumed that the initial equilibrium system satisfies the zero order equations. The linearized
continuity, energy and momentum equation become
[γ0ρ1 + γ1ρ0] +∇ · [γ0ρ1u+ γ0ρ0v1 + γ1ρ0u] = 0 , (A4)
γ20W1 − P1/c2 + 2γ40
u · v1/c2
γ20W1u+ 2γ
u · v1/c2
W0u+ γ
0W0v1
u · v1/c2
+ (1 + u2/c2)B0·B1 − (u ·B1/c+ v1·B0/c)u ·B0/c
2(B0·B1)u+B20v1 − (u ·B0)B1 − (u ·B1)B0 − (v1·B0)B0
= 0 ,
γ20W0
+ u · ∇v1
= −∇P1 −
(j0×B1) + (j1×B0)
. (A6)
The linearized Maxwell equations become:
∇ ·B1 = 0 ∇ ·E1 = 4πρq1
∇×B1 = 1c
j1 ∇×E1 = −1c
– 23 –
where I keep the displacement current in order to allow for strong magnetic fields and Alfvén wave
speeds comparable to lightspeed. Under the assumption of ideal MHD, the comoving electric field
is zero, the equilibrium charge density ρq,0 = 0, and the electric field
E1= −
u×B1 + v1×B0
is first order, the charge density ρq1 = (▽ ·E1) /4π is also first order, and the electrostatic force
term, ρq1E1, is second order and dropped from the linearized momentum equation. The condition
for isentropic perturbations becomes
P1 = ã
2ρ1 =
This basic set of linearized RMHD equations is similar to those found in Begelman (1998) but
allows a relativistic zeroth order velocity, i.e., v = u + v1 and u . c whereas Begelman allowed
only for relativistic first order motions, v1.
In what follows let us model a jet as a cylinder of radius R, having a uniform proper density,
ρj , a uniform axial magnetic field, Bj = Bz,j, and a uniform velocity, uj = uz,j. The external
medium is assumed to have a uniform proper density, ρe, a uniform axial magnetic field, Be = Bz,e,
and a uniform velocity, ue = uz,e. An external velocity could be the result of a wind or sheath
outflow around a central jet, ue > 0, or could represent backflow, ue < 0, in a cocoon surrounding
the jet. The jet is established in static total pressure balance with the external medium where the
total static uniform pressure is P ∗e ≡ Pe + B2e/8π = P ∗j ≡ Pj + B2j /8π. Under these assumptions
the linearized continuity equation becomes
[γ0ρ1 + γ1ρ0] + u
[γ0ρ1 + γ1ρ0] + γ0ρ0∇ · v1 = 0 . (A7)
The linearized energy equation becomes
γ20W1 −
+ 2γ40
γ20W1 + 2γ
+ γ20W0∇ · v1 = 0 . (A8)
This result for the linearized energy equation is found by noting that the magnetic terms in the
energy equation linearize to
Bz1 + u
− (uB0)∇ ·B1 +B20∇ · v1 −B20 ∂∂zvz1 =
∇ · v1 − ∂∂zvz1
∇ · v1 − ∂∂zvz1
where I have used
Bz1 + u
Bz1 = −
(rvr1) +
= −B0
∇ · v1 −
from ∂B1/∂t = ∇× (u×B1) +∇× (v1×B0). The linearized momentum equation becomes
γ20W0
v1 + u · ∇v1 − 14πc2γ2
v1×B0
−▽P1 − uc2
[(▽×B0)×B1 + (▽×B1)×B0] + 14πc2
– 24 –
where I have used
j0×B1
(∇×B0)×B1
j1×B0
(∇×B1)×B0
E1×B0 =
(∇×B1)×B0
v1×B0
×B0 ,
which includes the displacement current. The components of the linearized momentum equation
can be written as
γ20W0
vr1 + u
Br1 +
Br1 −
, (A10a)
γ20W0
vφ1 + u
Bφ1 +
Bφ1 −
(A10b)
γ20W0
vz1 + u
= − ∂
P1 (A10c)
where V 2A ≡ B20/(4πW0).
B. Normal Mode Dispersion Relation
In cylindrical geometry perturbations ρ1, v1, P1, andB1 can be considered to consist of Fourier
components of the form
f1(r, φ, z, t) = f1(r)e
i(kz±nφ−ωt)
where the flow is in the z direction, and r is in the radial direction with the jet bounded by r = R.
In cylindrical geometry n, an integer, is the azimuthal wavenumber, for n > 0 waves propagate at
an angle to the flow direction, where +n and −n refer to wave propagation in the clockwise and
counterclockwise sense, respectively, when viewed outwards along the flow direction. In general the
goal is to write a differential equation for the radial dependence of the total pressure perturbation
P ∗1 ≡ P1 + (B1 · B0)/4π = P ∗1 (r) exp[i(kz ± nφ − ωt)]. The differential equation can be obtained
from the energy equation by using the momentum equation and writing the velocity components
vr1, vφ1, vz1 in terms of P
1 , u,B0. The components of the linearized momentum equation (eqs.
A10a, b, & c) written in the form
γ20W0
vr1 + u
= − ∂
P ∗1 +
Br1 +
, (B1a)
γ20W0
vφ1 + u
P ∗1 +
Bφ1 +
, (B1b)
γ20W0
vz1 + u
P ∗1 +
Bz1 +
(B1c)
along with
B1 = −c(∇×E1) = ∇×(u×B1) +∇×(v1×B0)
– 25 –
are used to provide relations between Br1 and vr1, and Bφ1 and vφ1
Br1 + u
Br1 = B0
vr1 , (B2a)
Bφ1 + u
Bφ1 = B0
vφ1 , (B2b)
and to provide a relation between Bz1, vz1, and P
Bz1 + u
Bz1 =
vz1 + 2B0γ
vz1 + u
P ∗1 + u
P ∗1 .
(B2c)
To obtain equation (B2c) I have used
Bz1 + u
Bz1 = −
(rvr1) +
= −B0
▽ · v1 −
where
−γ20W0▽ · v1 = γ20
P1 + u
P1 + 2γ
vz1 + u
from the energy equation (eq. A8), and
W1 = ρ1 +
Using equations (B1a, b, & c) combined with
f1(r, φ, z, t) = −iωf1(r)ei(kz±nφ−ωt)
f1(r, φ, z, t) =
f1(r)e
i(kz±nφ−ωt)
f1(r, φ, z, t) = ±inf1(r)ei(kz±nφ−ωt)
f1(r, φ, z, t) = +ikf1(r)e
i(kz±nφ−ωt)
allows the velocity components to be written as
iγ20W0
ku− ω
vr1 = −
P ∗1 + i
k − ω u
Br1 , (B3a)
iγ20W0
ku− ω
vφ1 = −
P ∗1 + i
k − ω
Bφ1 , (B3b)
γ20W0 [ku− ω] vz1 = −
k − ω
P ∗1 +
k − ω
Bz1 . (B3c)
– 26 –
The perturbed magnetic field components from equations (B2a, b,& c) become
Br1 =
ku− ω
B0 , (B4a)
Bφ1 =
ku− ω
B0 , (B4b)
Bz1 =
(ku−ω)
k − ωu/c2
P ∗1 + γ
k − ωu/c2
+ (ku− ω) u
(ku− ω) + V 2A
(ku−ω)
− (k − ωu/c2) u
] B0 (B4c)
where I have used
k + 2γ20 (ku− ω)u/c2 = γ20
k − ωu/c2
+ (ku− ω)u/c2
γ20 (ku− ω)
ã−2 + Γ (Γ− 1)−1 c−2
+ ω/c2 = γ20
(ku− ω) /a2 −
k − ωu/c2
to obtain the expression for Bz1. Using equations (B4a, b, & c) for the perturbed magnetic field
components, I obtain the following relations between the perturbed velocity components v1 and
the total pressure perturbation P ∗1 :
vr1 ≡ Cr
P ∗1 = i
P ∗1 = i
(ku− ω)
γ20W0γ
(ku− ω)2 − (k − ωu/c2)2 v2A
P ∗1 , (B5a)
vφ1 ≡ CφP ∗1 = ∓
P ∗1 = ∓
(ku− ω)
γ20W0γ
(ku− ω)2 − (k − ωu/c2)2 v2A
]P ∗1 , (B5b)
vz1 ≡ CzP ∗1 = −
(ku− ω)
k − ωu/c2
γ20W0
(ku− ω)2 + γ2Av2A
(ku−ω)2
− (k − ωu/c2)2
]}P ∗1 . (B5c)
To obtain the above relationships I have used
(ku− ω)−
k − ωu/c2
ku− ω
= γ2A
(ku− ω)−
k − ωu/c2
(ku− ω)
in addition to
γ20 (ku− ω)
ã−2 + Γ (Γ− 1)−1 c−2
+ ω/c2 = γ20
(ku− ω) /a2 −
k − ωu/c2
where
v2A ≡
1 + V 2A/c
– 27 –
is the Alfvén wave speed and γ2A =
1− v2A/c2
is an Alfvénic Lorentz factor. Note that V 2A =
A and γ
A = 1 + V
2. Thus we have that
∇ · v1 = Cr ∂
P ∗1 +
P ∗1 +
P ∗1 + Cz
P ∗1 +
P ∗1 − n
P ∗1 + ikCzP
Using the energy equation (eq. A8) written in the form
−γ20W0▽ · v1 = γ20
P ∗1 + u
Bz1 + u
Bz1 + 2γ
vz1 + u
inserting
Bz1 + u
Bz1 = −B0
▽ · v1 −
and using vz1 = CzP
1 gives
∇ · v1 = −i Yγ2
(ku−ω)
(ku−ω)
]−1 [
2γ20 (ku− ω) uc2 −
(ku−ω)
where
Y = γ20
(ku− ω) /a2 −
k − ωu/c2
. (B8)
Setting equations (B6) and (B7) equal gives us a differential equation for P ∗1 in the form of Bessel’s
equation
P ∗1 + r
P ∗1 +
β2r2 − n2
P ∗1 = 0 (B9)
where
β2 = Y X
(ku−ω)
+kXCz +
(ku−ω)
]−1 [
2γ20 (ku− ω) uc2 −
(ku−ω)
XCz .
(B10)
I can simplify the expression for β2 by writing
(ku− ω) + V
(ku− ω)
γ20W0
(ku− ω) +
kCz +
2γ20 (ku− ω)
kY Cz
from which it follows that
(ku− ω) + V
(ku− ω)
γ20W0
Y + γ20 (ku− ω)
k − ωu/c2
+ (ku− ω) u/c2
– 28 –
where I have used 2γ20 (ku− ω)u/c2 = γ20
k − ωu/c2
+ (ku− ω)u/c2
− k. Substituting the ex-
pressions for X and Cz from equations (B5a, b, & c), and Y from equation (B8) and modest
algebraic manipulation yields
(ku−ω)2−(k−ωu/c2)
(a2+γ2Av
A)(ku−ω)
a2(ku−ω)(k−ωu/c2)u/c2
(ku− ω)2 + (ku− ω)
k − ωu/c2
a2u/c2 −
(ku−ω)2
(k−ωu/c2)
a2+(ku−ω)(k−ωu/c2)ua2/c2
(a2+γ2Av
A)(ku−ω)
a2(k−ωu/c2)
(B11)
Additional regrouping provides the following form
(ku−ω)2−(k−ωu/c2)
(a2+γ2Av
A)(ku−ω)
a2(ku−ω)(k−ωu/c2)u/c2
[(a2+γ2Av
A)(ku−ω)
a2(ku−ω)(k−ωu/c2)u/c2]
(ku−ω)2−(k−ωu/c2)
(a2+γ2Av
A)(ku−ω)
a2(k−ωu/c2)
} (B12)
from which I find that β2 can be written in the compact form:
̟2 − κ2a2
̟2 − κ2v2A
v2ms̟
2 − κ2v2Aa2
. (B13)
where ̟2 ≡ (ω − ku)2, κ2 ≡
k − ωu/c2
, and where the fast magnetosonic speed perpendicular
to the magentic field is given by (e.g., Vlahakis & Königl 2003)
vms ≡
a2 + v2A − a2v2A/c2
a2/γ2A + v
It is easily seen that this expression for β2 reduces to the relativistic pure fluid form
β2 −→
̟2 − κ2a2
= γ20
(ku− ω)2
k − ωu/c2
given in Hardee (2000) and that this expression for β2 reduces to the non-relativistic MHD form
β2 −→
̟2 − κ2a2
̟2 − κ2v2A
a2 + v2A
̟2 − κ2v2Aa2
(ku− ω)4
a2 + V 2A
(ku− ω)2 − k2V 2Aa2
where κ2 −→ k2 and v2A −→ V 2A given in Hardee, Clarke & Rosen (1997).
The solutions that are well behaved at jet center and at infinity are P ∗j1(r ≤ R) = CjJ±n(βjr),
and P ∗e1(r ≥ R) = CeH
±n(βer), respectively, where J±n and H
±n are the Bessel and Hankel
functions with arguments defined as
β2j ≡
̟2j − κ2ja2j
̟2j − κ2jv2Aj
v2msj̟
j − κ2jv2Aja2j
 , (B14a)
– 29 –
β2e ≡
̟2e − κ2ea2e
̟2e − κ2ev2Ae
v2mse̟
e − κ2ev2Aea2e
, (B14b)
where̟2j,e ≡ (ω − kuj,e)
, κ2j,e ≡
k − ωuj,e/c2
, γ2j,e ≡
1− u2j,e/c2
and γ2Aj,e ≡
1− v2Aj,e/c2
The jet flow speed and external flow speed are positive if flow is in the +z direction.
The condition that the total pressure be continuous across the jet boundary requires that
CjJ±n(βjR) = CeH
±n(βeR) . (B15)
The first derivative of the total pressure is given by
P ∗1 = −iXvr1 .
and with
vr1 ≡
+ u · ∇
ξr = −i (ω − ku) ξr
where ξr is the fluid displacement in the radial direction it follows that
∂P ∗1
= − (ω − ku)Xξr . (B16)
The radial displacement of the jet and external medium must be equal at the jet boundary, i.e.,
r(R) = ξ
r(R), from which it follows that
− (ω − kuj)Xj
∂Jn(βjr)
∂ (βjr)
− (ω − kue)Xe
n (βer)
∂ (βer)
. (B17)
Inserting Cj and Ce in terms of the Bessel and Hankel functions leads to a dispersion relation
describing the propagation of Fourier components which can be written in the following form:
n(βjR)
Jn(βjR)
n (βeR)
n (βeR)
. (B18)
where the primes denote derivatives of the Bessel and Hankel functions with respect to their argu-
ments. The expressions
χj ≡ γ2j γ2AjWj
̟2j − κ2jv2Aj
(B19a)
χe ≡ γ2eγ2AeWe
̟2e − κ2ev2Ae
(B19b)
readily reduce to the non-relativistic form χ = ρ0[(ω − ku)2 − k2V 2A] where W0 −→ ρ0 given in
Hardee, Clarke & Rosen (1997). This dispersion relation describes the normal modes of a cylindrical
jet where n = 0, 1, 2, 3, 4, etc. involve pinching, helical, elliptical, triangular, rectangular, etc.
normal mode distortions of the jet, respectively.
– 30 –
C. Analytic Solutions and Approximations
Each normal mode n contains a fundamental/surface wave and multiple body wave solutions
to the dispersion relation. The low-frequency limiting form for the fundamental/surface modes are
obtained in the limit where ω −→ 0 and k −→ 0 but with ω/k 6= 0. In this limit the dispersion
relation for the fundamental (n = 0) and surface (n > 0) modes is given by
χj ≈ −12χe (βjR)
) + π
ǫ− iπ
n = 0 (C1)
χj ≈ −χe n > 0 (C2)
where in this limit βeR −→ 0 and βjR −→ 0, and I have used the small argument forms for the
Bessel and Hankel functions to write
n(βjR)
Jn(βjR)
n (βeR)
n (βeR)
(βeR) (βjR)
) + π
ǫ− iπ
n = 0
−βe/βj n > 0
where ǫ is Euler’s constant.
C.1. Fundamental Pinch Mode (n = 0 ; m = 0) in the low frequency limit
In the low frequency limit, dispersion relation solutions for the fundamental axisymmetric
pinch mode are obtained from equation (C1)
γ2j γ
̟2j − κ2jv2Aj
̟2e − κ2ev2Ae
) + π
ǫ− iπ
Here we have the trivial solution ̟2j − κ2jv2Aj = 0 with v2w = v2Aj and the more interesting zeroth
order solution
̟2j ≈ κ2j
v2Aja
v2msj
with wave speed in the proper frame given by
v2w = ̟
v2Aja
v2msj
. (C5)
To first order this magnetosonic wave solution (eq. C3) can be written as
̟2j [1− δ] ≃ κ2ja2j
v2msj
, (C6)
– 31 –
where
δ ≡ −
̟2e − κ2ev2Ae
v2msj
and δ is complex. Thus, in the low frequency limit this fundamental pinch mode (n = 0) solution
consists of a growing and damped wave pair with wave speed in the observer frame
uj ± vw
1± vwuj/c2
where
v2w ≃ a2j
v2msj
v2msj
. (C9)
Previous work has shown the unstable growing solution associated with the backwards moving (in
the jet fluid reference frame) wave.
C.2. Surface Modes (n > 0 ; m = 0) in the low frequency limit
In the low frequency limit the fundamental dispersion relation solution for all higher order
modes (n > 0) is most easily obtained from equation (C2) written in the form
γ2jWj
(ω − kuj)2 −
V 2Aj
k2 − ω2/c2
= −γ2eWe
(ω − kue)2 −
V 2Ae
k2 − ω2/c2
(C10)
where I have used χ ≡ γ20γ2AW0
̟2 − κ2v2A
= γ20W0
(ω − ku)2 −
k2 − ω2/c2
V 2A/γ
. The solu-
tion can be put in the form
[ηuj + ue]± iη1/2
(uj − ue)2 − V 2As/γ2j γ2e
(1 + V 2Ae/γ
2) + η(1 + V 2Aj/γ
(C11)
where
γ2jWj
γ2eWe
and a “surface” Alfvén speed is defined by
V 2As ≡
γ2AjWj + γ
) B2j +B
4πWjWe
The jet is stable to higher order n > 0 fundamental mode perturbations when
γ2j γ
e (uj − ue)
< γ2Ajγ
Ae +We/γ
) B2j +B
4πWjWe
. (C12)
Equation (C11) reduces to the relativistic fluid expression
ηuj + ue
1 + η
1 + η
(uj − ue) (C13a)
– 32 –
given in Hardee & Hughes (2003) equation (6a) where for pressure balance and equal adiabatic
index in jet and external medium η −→ γ2j ae/γ2eaj . Similarly equation (C11) reduces to the
non-relativistic MHD form
ηuj + ux
1 + η
1 + η
(uj − ue)2 − V 2As
(C13b)
given by Hardee & Rosen (2002) eq. (4) where V 2As −→ (ρj + ρe)
B2j +B
/ (4πρjρe) and η −→
ρj/ρe.
C.3. Body Modes (n ≥ 0 ; m ≥ 1) in the low frequency limit
In the low frequency limit the real part of the body wave solutions can be obtained in the limit
ω = 0, k 6= 0 where the dispersion relation can be written in the form
cos [βjR− (2n + 1)π/4] ≈ ǫn ≡
n(βjR)
n (βeR)
n (βeR)
. (C14)
Here I have assumed that the large argument form Jn(βjR) ≈ (2/πβjR)1/2 cos [βjR− (2n + 1)π/4]
applies. In the absence of a magnetic field and a flow surrounding the jet, χe = 0, ǫn = 0, and
solutions are found from βjR − (2n + 1)π/4 = ±mπ ± π/2, where m is an integer. Provided
ǫn << π/2 and θ ≈ cos−1 ǫn ≈ ± (π/2− ǫn), solutions can be found from βjR − (2n + 1)π/4 =
± [mπ + (π/2 ± ǫn)], where for ±ǫn the plus or minus sign is for m odd or even, respectively. In
the limit ω = 0
βjR ≈
γ2j (u
j − a2j)(u2j − v2Aj)
v2msju
j − v2Aja2j
kR , (C15)
and the solutions are given by
kR ≈ kminnmR ≡
v2msju
j − v2Aja2j
γ2j (u
j − a2j )(u2j − v2Aj)
× [(n+ 2m− 1/2)π/2 + (−1)mǫn] (C16)
where I have set m −→ m+1 to be consistent with previous notation so m = 1 corresponds to the
first body mode.
In the limit a2j >> v
Aj equation (C16) reduces to the relativistic purely fluid form found in
Hardee & Hughes (2003)
kR ≈ kminnmR ≡
[(n+ 2m− 1/2)π/2 + (−1)mǫn]
M2j − 1
(C17a)
where M2j = u
j . In the limit v
Aj >> a
j equation (C16) becomes
kR ≈ kminnmR ≡
[(n+ 2m− 1/2)π/2 + (−1)mǫn]
M2Aj − 1
(C17b)
where M2Aj = u
– 33 –
Equation (C16) reduces to the non-relativistic MHD form found in Hardee & Rosen (2002)
kR ≈ kminnmR ≡
1−M2ms/M2AjM2j
× [(n+ 2m− 1/2)π/2 + (−1)mǫn] (C17c)
where M2ms = u
j + v
Aj) and I have used
γ2j (u
j − a2j)(u2j − v2Aj)
v2msju
j − v2Aja2j
= γ2j
a2+v2
1−M2ms/M2AjM2j
a2+v2
We note here that there is an error in equations (5) in previous articles in the treatment of the sign
on ǫn for even values of m.
C.4. The Resonance Condition
The resonance conditions are found by evaluating the transmittance, T , and reflectance, R, of
waves at the jet boundary where T = 1 +R. With the dispersion relation written as
n(βjR)
Jn(βjR)
n (βeR)
n (βeR)
(C18)
where Z = χ/β with
Z = γ2γ2AW
̟2 − κ2v2A
a2 + γ2Av
̟2 − γ2Aκ2v2Aa2
γ2γ2A (̟
2 − κ2a2)
̟2 − κ2v2A
(C19)
the reflectance
R = (Ze − Zj)/(Ze + Zj) . (C20)
For a fluid containing no magnetic field Z is a quantity related to the acoustic normal impedance
(Gill 1965). When Ze + Zj ≈ 0, R and T are large, and the reflected and transmitted waves have
an amplitude much larger than the incident wave.
C.4.1. The Fluid Limit (Alfvén speed ≪ sound speed)
For the case of a pure fluid
ζ2e + γ
, (C21a)
ζ2j + γ
, (C21b)
where χ/k2 = W
ζ2 + γ2κ2/k2
a2 and ζ ≡ β/k. For non-relativistic flows where (u2/c2)(ω/ku) <<
1, γ ≈ 1, and with adiabatic indicies Γj = Γe the reflectance
(ζe − ζj)(ζeζj − 1)
(ζe + ζj)(ζeζj + 1)
(C22)
– 34 –
and a supersonic resonance (Miles 1957) occurs when βe + βj = k(ζe + ζj) = 0. This supersonic
resonance corresponds to the maximum growth rate of the normal mode solutions to the dispersion
relation.
I now generalize the results in Hardee (2000) to include flow in the external medium relative
to the source/observer frame. Here Ze + Zj = 0 becomes
Γeζjχe + Γjζeχj = Γeζj
+ Γjζe
= 0 . (C23)
A necessary condition for resonance is ζj < 0 and ζe > 0, and on the real axis
uj − aj
1− ujaj/c2
ue + ae
1 + ueae/c2
. (C24)
It follows that the resonance only exists when
uj − aj
1− ujaj/c2
ue + ae
1 + ueae/c2
(C25a)
or equivalently
uj − ue
1− ujue/c2
aj + ae
1 + ajae/c2
. (C25b)
To find the resonant solution for the real part of the phase velocity I solve ζ2j = ε
2ζ2e where
here I set ε ≡ (Γjγ2j̟2j/k2a2j )/
= 1 so that
ζ2j = γ
(ω/k − uj)2
= ζ2e = γ
(ω/k − ue)2
. (C26)
The resulting quadratic equation can be written in the form
a2e/(γ
sj)− a2j/(γ2j γ2se)
)2 − 2
γ2j a
euj/γ
sj − γ2ea2jue/γ2se
γ2j a
u2j − a2j
− γ2ea2j
u2e − a2e
= 0 ,
(C27)
where I have used
γ2j a
− γ2ea2j
a2e/(γ
sj)− a2j/(γ2j γ2se)
and γ2s ≡
1− a2/c2
. The solutions to equation (C27) are given by
γ2j a
euj/γ
sj − γ2ea2jue/γ2se
a2e/(γ
sj)− a2j/(γ2j γ2se)
γseγsj
u2j − 2ujue + u2e
a2e/(γ
sj)− a2j/(γ2j γ2se)
] (C28)
with the resonant solution given by
v∗w =
(γseae)γjuj + (γsjaj)γeue
γj(γseae) + γe(γsjaj)
. (C29)
– 35 –
Inserting the resonant solution (eq. C29) into the expression for ε gives
0.695 ≤ ε2 =
≤ 1.44
where 2.78 ≤ Γ2γ4s ≤ 4. When aj = ae and Γj = Γe, ε2 = 1, and the resonant solution is exact. The
small range on ε (0.83 ≤ ε ≤ 1.2) suggests that this solution remains relatively robust for unequal
values of the sound speed and adiabatic index in the jet and external medium. In the absence of
an external flow the resonant solution
v∗w =
(γseae)γjuj
γj(γseae) + (γsjaj)
M2j − β2
M2j − β2
+ (M2e − β2)
is equivalent to the form given in Hardee (2000).
The resonant frequencies can be estimated using the large argument forms for the Bessel and
Hankel functions. In this limit the dispersion relation becomes
n(βjR)H
n (βeR)
Jn(βjR)H
n (βeR)
≈ i tan(βjR−
2n+ 1
. (C30)
From the dispersion relation with Ze + Zj ≈ 0, and (χj/βj)(βe/χe) = Zj/Ze ≈ βe/βj ≈ −1,
tan[βjR− (2n + 1)π/4]Re ≈ 0 on the real axis. It follows that
|βjR| ≈ |βeR| ≈ (2n+ 1)π/4 +mπ
can be used to obtain an estimate for the resonant frequencies from |βeR| ≈ (2n + 1)π/4 + mπ,
with result that the resonant frequencies are given by
ω∗nmR
(2n + 1)π/4 +mπ
(1− ue/v∗w)
2 − (ae/v∗w − ueae/c2)
. (C31)
In the absence of external flow, ue = 0, and for uj >> ae and 1 >> (kae/ω)
2 this expression
reduces to the form given in Hardee (2000).
When γj(γseae) >> γe(γsjaj), the resonant wave speed becomes v
w ≈ uj, ue/v∗w ≈ ue/uj and
provided ue << uj and ae << uj , the resonant frequency increases with increasing ue/uj and ae/uj
ω∗nmR
(2n + 1)π/4 +mπ
1− 2ue/uj(1− a2e/c2)− (a2e − u2e)/u2j
. (C32)
In general, the resonant frequency ω∗nm −→ ∞ as (1− ue/v∗w)
w − ueae/c2
)2 −→ 0. An
equivalent condition for (1− ue/v∗w)
w − ueae/c2
= 0 is
uj − ue
1− ujue/c2
aj + ae
1 + ajae/c2
, (C33)
and the resonance moves to higher frequencies with ω∗nm −→ ∞ when the“shear” speed drops below
a “surface” sound speed.
– 36 –
The behavior of the growth rate at resonance also can be found using the large argument forms
for the Bessel and Hankel functions. In this limit the reflectance can be written as
(Ze − Zj)
(Ze + Zj)
Jn(βjR)H
n (βeR)− J
n(βjR)H
n (βeR)
Jn(βjR)H
n (βeR) + J
n(βjR)H
n (βeR)
≈ exp[−2i(βjR−
2n+ 1
π)] , (C34)
2n+ 1
π ≈ i
ln |R| − φ
(C35)
where R ≡ |R| eiφ. It follows that
(βjR)I ≈
ln |R| (C36)
and since typically at resonance, |ω − kRuj | /aj >
∣kR − ωuj/c2
∣ I can approximate βj by
βj ≡ βRj + iβIj ≈ γj
(ω − kRuj)
− ikI
. (C37)
It follows that
(βjR)I ≈ −γj
kIR , (C38)
kIR ≈ −
ln |R| . (C39)
At resonance
(Ze − Zj)
(Ze + Zj)
βj − βe(χj/χe)
βj + βe(χj/χe)
βj − βe
βj + βe
≈ −2βe
βj + βe
(C40)
|R| ≈
βj − βe
βj + βe
−2βRe
βIj − βIe
βIj + β
(C41)
where I have used
βRj − βRe
≈ −2βRe from the resonance condition on the real axis. It follows
|R| ≈
(ω−kRue)
+ k2I
− γe ueae
− γeae
ω−kRue
ω−kRue
(C42)
where I have used
βe ≡ βRe + iβIe ≈ γe
(ω − kRue)
− ikI
ω − kRue
If I assume that γj(γseae) >> γe(γsjaj), with resonant wave speed ω/kR ≈ uj and ue/uj << 1
|R| ≈
ω∗nmR
(1− 2ue/uj) + k2IR2
(1 + ue/uj)
(1 + ue/uj)
, (C43)
– 37 –
and since kIR ≈ − (aj/2γjuj) ln |R|
|R| ≈
ω∗nmR
(1− 2ue/uj) + [ln |R| /2]2
[ln |R| /2]2
. (C44)
From equation (C32)
ω∗nmR
(1− 2ue/uj) ≈
(1− 2ue/uj)
1− 2 (ue/uj) (1− a2e/c2)− a2e/u2j
] [(2n+ 1)π/4 +mπ]
and if say ue = 0, then
|R|2 − 1
ln |R| ≈ 4
1− a2e/u2j
[(2n + 1)π/4 +mπ] . (C45)
Formally |R| −→ ∞ as ω∗nm −→ ∞ when the jet speed drops below the “surface” sound speed
given by equation (C33). This result applies only to the surface modes and not to the body modes
as, in the fluid limit, the body modes do not exist when the jet speed drops below the jet sound
speed, see equation (C17). On the other hand, if say, a2e/u
j << 1, then
|R|2 − 1
ln |R| ≈ 4 [(2n + 1)π/4 +mπ] . (C46)
Formally |R| ≈ constant as ω∗nm −→ ∞ when the wind speed becomes comparable to the jet speed,
ue . uj , as must be the case for the velocity shear driven Kelvin-Helmholtz instability.
C.4.2. The Magnetic Limit (Alfvén speed ≫ sound speed)
For the magnetic limit in which magnetic pressure dominates gas pressure
Ze = γeγ
AeWevAe
̟2e − κ2ev2Ae
, (C47a)
Zj = γjγ
AjWjvAj
̟2j − κ2jv2Aj
, (C47b)
A necessary condition for resonance is
̟2e − κ2ev2Ae
> 0 and
̟2j − κ2jv2Aj
< 0 on the real axis
with result that Ze + Zj = 0 when
uj − vAj
1− ujvAj/c2
ue + vAe
1 + uevAe/c2
. (C48)
It follows that the resonance only exists when
uj − ue
1− ujue/c2
vAj + vAe
1 + vAjvAe/c2
. (C49)
This result is identical in form to the sonic case with sound speeds replaced by Alfvén wave speeds.
– 38 –
The resonant solution for the real part of the phase velocity is obtained from
Z2j = γ
̟2j −
k2 − ω2/c2
V 2Aj/γ
= Z2e = γ
̟2e −
k2 − ω2/c2
V 2Ae/γ
(C50)
where I have used γ2γ2A
̟2 − κ2v2A
k2 − ω2/c2
V 2A/γ
, and recall that v2A = V
The resulting quadratic equation can be written in the form
Aj − γ2eW 2e V 2Ae
)2 − 2
Ajuj − γ2eW 2e V 2Aeue
j − γ2eW 2e V 2Aeu2e
(C51)
where I have used
k2 − ω2/c2
V 2Aj/γ
j = γ
k2 − ω2/c2
V 2Ae/γ
because pressure balance in the magnetically dominated case requires WjV
Aj = WeV
Ae. The
solutions are given by
Ajuj − γ2eW 2e V 2Aeue ± γjγeWjWeVAjVAe (uj − ue)
Aj − γ2eW 2e V 2Ae
, (C52)
and the resonant solution becomes
v∗w =
γjWjVAjuj + γeWeVAeue
γjWjVAj + γeWeVAe
(γAevAe) γjuj + (γAjvAj) γeue
γj (γAevAe) + γe (γAjvAj)
(C53)
where I have used WVA = WV
A/ (γAvA), and WjV
Aj = WeV
Ae. This resonant solution has the
same form as the sonic case with sound speeds and sonic Lorentz factors replaced by Alfvén wave
speeds and Alfvénic Lorentz factors.
As in the sonic case the resonant frequencies are found from
|βeR| ≈ (2n + 1)π/4 +mπ
with result that the resonant frequencies are given by
ω∗nmR
(2n + 1)π/4 +mπ
(1− ue/v∗w)
2 − (vAe/v∗w − uevAe/c2)
. (C54)
When γj (γAevAe) >> γe (γAjvAj) the resonant wave speed becomes v
w ≈ uj and ue/v∗w ≈ ue/uj ,
and provided ue << uj and vAe << uj the resonant frequency increases with increasing ue/uj and
vAe/uj as
ω∗nmR
≈ (2n + 1)π/4 +mπ
1− 2 (ue/uj) (1− v2Ae/c2)− (v2Ae − u2e)/u2j
. (C55)
Here the resonant frequency ω∗nm −→ ∞ as (1− ue/v∗w)
vAe/v
w − uevAe/c2
)2 −→ 0. An
equivalent condition for (1− ue/v∗w)
vAe/v
w − uevAe/c2
= 0 is
uj − ue
1− ujue/c2
vAj + vAe
1 + vAjvAe/c2
, (C56)
– 39 –
and the resonance moves to higher frequencies as the “shear” speed becomes trans-Alfvénic.
The behavior of the growth rate at resonance proceeds in the same manner as for the fluid
limit but with sound speeds replaced by Alfvén wave speeds. The resonant growth rate is now
given by
kIR ≈ −
ln |R| . (C57)
If I assume that γj(γAevAe) >> γe(γAjvAj), with resonant wave speed ω/kR ≈ uj and ue/uj << 1
|R| ≈
ω∗nmR
(1− 2ue/uj) + k2IR2
− vAe
(1 + ue/uj)
+ vAe
(1 + ue/uj)
, (C58)
From equation (C54)
ω∗nmR
(1− 2ue/uj) ≈
(1− 2ue/uj)
1− 2 (ue/uj) (1− v2Ae/c2)− (v2Ae − u2e)/u2j
] [(2n+ 1)π/4 +mπ]
and if say ue = 0, then
|R|2 − 1
ln |R| ≈ 4
1− v2Ae/u2j
[(2n + 1)π/4 +mπ] (C59)
and |R| increases as ω∗nm increases when the jet speed decreases. However, when the shear speed
drops below the “surface” Alfvén speed, see equations (C11 & C12) the jet is stable. This result
is quite different from the fluid limit where the jet remains unstable when the shear speed drops
below the “surface” sound speed. If I insert
uj − ue =
1− ujue/c2
1 + vAjvAe/c2
(vAj + vAe) .
from equation (C56) into equation (C12), it follows that the jet will be unstable when resonance
disappears only when
γ2j γ
1− ujue/c2
> 2γ2Ajγ
v2Ae + v
(vAj + vAe)
1 + vAjvAe/c
. (C60)
where I have used
v2Ae + v
Ae +We/γ
B2j +B
/ (4πWjWe)
as Be = Bj from
magnetic pressure balance. Formally |R| −→ ∞ as ω∗nm −→ ∞ only for jet Lorentz factors greatly
in excess of the Alfvénic Lorentz factor.
C.5. Wave modes at high frequency
To obtain the behavior of wave modes at high frequency I begin with the dispersion relation
written in the form
Jn(βjR)
n(βjR)
n (βeR)
n (βeR)
Jn(βjR)
∓Jn±1(βjR)± nβjRJn(βjR)
n−1(βeR)−
n (βeR)
n (βeR)
(C61)
– 40 –
and assume a large argument in the Hankel function with H
n (βeR) ≈ exp i [βeR− (2n+ 1) π/4]
and a small argument βjR << 1 in the Bessel function to write
J0(βjR)
−J1(βjR)
e−iπ/2 n = 0
e−iπ/2 n > 0
The small arguement form for the Bessel function gives J0(βjR)/J1(βjR) ≈ 2/βjR with result that
the dispersion relation becomes
βjR ≈
e−iπ/2
n = 0
e−iπ/2 n > 0
. (C62)
At high frequency and large wavenumber χj and χe, are proportional to k
2, βj and βe are pro-
portional to k, and βjR = ζjkR ∝ (kR)1/2 for n = 0. Thus, the internal solutions in the high
frequency and large wavenumber limit are given by βjR ≃ 0 and are found from
(kuj − ω)2 −
k − ωuj/c2
(kuj − ω)2 −
k − ωuj/c2
≈ 0 (C63)
and it follows that
uj ± aj
1± ujaj/c2
, (C64a)
uj ± vAj
1± ujvAj/c2
. (C64b)
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This preprint was prepared with the AAS LATEX macros v5.2.
http://arxiv.org/abs/astro-ph/0609254
http://arxiv.org/abs/astro-ph/0611406
	Introduction
	The RMHD Normal Mode Dispersion Relation
	Analytical Solutions to the Dispersion Relation
	Low Frequency Limit
	Resonance
	High Frequency Limit
	Numerical Solution of the Dispersion Relation
	Fluid Limit
	Magnetic Limit
	A High Sound and Alfvén Speed Magnetosonic Case
	Summary
	Discussion
	Linearization of the RMHD Equations
	Normal Mode Dispersion Relation
	Analytic Solutions and Approximations
	Fundamental Pinch Mode (n = 0 ; m = 0) in the low frequency limit
	Surface Modes (n > 0 ; m = 0) in the low frequency limit
	Body Modes (n 0 ; m 1) in the low frequency limit
	The Resonance Condition
	The Fluid Limit (Alfvén speed  sound speed)
	The Magnetic Limit (Alfvén speed  sound speed)
	Wave modes at high frequency
