The Parker Transport Equation€¦ · often called the \transport equation" ASP2018. The Parker...

$: The Parker Transport Equation€¦ · often called the \transport equation" ASP2018. The Parker Transport Equation Riaan Steenkamp Current Density in the Solar Wind I Consider gas$
The ParkerTransportEquation

RiaanSteenkamp

The Parker Transport Equation

Riaan Steenkamp

2018

ASP2018


RiaanSteenkamp

Structure

Distribution functions & differential quantities

CR Continuity Equations

Current Density in the Solar Wind & Stationary frames

The Transport Equation

Diffusive Schock Acceleration

Solutions of the The Parker Transport Equation

Anomalous Component of Cosmic Rays

ASP2018


RiaanSteenkamp

Distribution functions & differential quantities I

Cosmic rays come from all directions in outer space overlarge energy intervals

Number of particles in volume element d3r about r in themomentum interval d3p about p is given by

dn = F (r ,p, t) d3r d3p

with F the distribution function.

I.t.o. a space angle dΩ about direction of p it is

dn = F (r ,p, t) d3r p2 dp dΩ

Can typically not measure F , but only averages overmomentum space.

ASP2018


RiaanSteenkamp

Distribution functions & differential quantities II

Average of F over all directions is

f (r , p, t) =1

4π

∫ΩF (r ,p, t) dΩ

with∫

Ω dΩ = 4π

Number of particles dN in d3r in (p, p + dp) (independentof direction of p) is

dN = 4πp2f (r , p, t) d3r dp.

Define the differential number density Up as the numberof particles in d3r about r with momentum in the interval(p, p + dp) with

dN = Up(r , p, t) d3r dp

ASP2018


RiaanSteenkamp

Distribution functions & differential quantities III

ThusUp = 4πp2f

with units of “particles per volume interval”.

Detectors do not measure Up, but rather differentialintensity.

If particle velocity has magnitude v , the differentialintensity is

jp = vUp

or “number of particles / unit momentum / unit surfacearea / unit time” that goes from ALL directions (i.e. 4πsteradian) through detector.

ASP2018


RiaanSteenkamp

Distribution functions & differential quantities IV

Usually normalised as w.r.t. unit space angle:

jp =vUp

4π

or “number of particles / unit momentum / unit surfacearea / steradian / unit time”

In practice we measure particles w.r.t. kinetic energyintervals and not momentum intervals:

From E 2 = p2c2 + E 20 we have that dT = v dp

ASP2018


RiaanSteenkamp

Distribution functions & differential quantities V

Since we specify same particles in two ways, we have thatjT dT = jp dp it follows that

jT = jpdp

dT=

1

vjp

Now we can summarise the following important relations

jT =Up

4π= p2f

IMPORTANT: jT measured by experimentalists has simplerelationship to quantities Up and f used by theorteticians!

ASP2018


RiaanSteenkamp

CR Continuity Equations I

Consider the distribution function F (r ,p, t). Now

dF

dt=∂F

∂t+ v ·∇F + p·∇pF

with ∇p the gradient operator w.r.t. to momentum space,v = (x , y , z) and p = (px , py , pz).

Integrate this over all directions in momentum space:∫Ω

dF

dtdΩ =

∫Ω

∂F

∂tdΩ +

∫Ωv ·∇F dΩ +

∫Ωp·∇pF dΩ

ASP2018


RiaanSteenkamp

CR Continuity Equations II

LHS: ∫Ω

dF

dtdΩ =

d

dt

∫ΩF dΩ = 4π

df

dt

RHS term 1:∫Ω

∂F

∂tdΩ =

∂

∂t

∫ΩF dΩ = 4π

∂f

∂t

Now

4πdf

dt= 4π

∂f

∂t+

∫Ωv ·∇F dΩ +

∫Ωp·∇pF dΩ

ASP2018


RiaanSteenkamp

CR Continuity Equations III

Evaluate∫

Ω v ·∇F dΩ:

Look at x-component:∫Ωvx∂F

∂xdΩ =

∫Ω

[∂

∂x(vxF )− ∂vx

∂xF

]dΩ

=∂

∂x

∫ΩvxF dΩ

But

〈vx〉 =

∫Ω vxF dΩ∫

Ω F dΩ=

1

4πf

∫ΩvxF dΩ

ASP2018


RiaanSteenkamp

CR Continuity Equations IV

Therefore

4πf 〈vx〉 =

∫ΩvxF dΩ

and ∫Ωvx∂F

∂xdΩ = 4π

∂

∂x(〈vx〉 f )

Therefore LHS term 2 becomes∫Ωv ·∇F dΩ = 4π∇·(〈v〉 f )

Now

4πdf

dt= 4π

∂f

∂t+ 4π∇·(〈v〉 f ) +

∫Ωp·∇pF dΩ

ASP2018


RiaanSteenkamp

CR Continuity Equations V

The last term is more involved:∫Ωp·∇pF dΩ =

∫Ω

[∇p·(pF )− F∇p·p] dΩ

=

∫Ω∇p·(pF ) dΩ

since ∇p·p = ∂px∂px

+∂py∂py

+ ∂pz∂pz

= 0,

because, while E-M forces F = p = q(E + p×B/m) arenot velocity independent, px is independent of px and thus∂px∂px

= 0, etc.

Use Divergence Theorem (∫∇·A d3x =

∫Enc.S A·da):∫

∇p(pF ) d3p =

∫Enc.S

(pF )·da =

∫Ω

(pF )·ep p2 dΩ

ASP2018


RiaanSteenkamp

CR Continuity Equations VI

and ∫∇p(pF ) d3p =

∫Ωp2pF dΩ

since p·(pep) = p·p = pp.

Let us integrate the above from p to p + dp:∫ p+dp

p∇p(pF ) d3p =

∫Ω,p+dp

p2pF dΩ−∫

Ω,pp2pF dΩ

= 4π[(p2 〈p〉 f

)p+dp

−(p2 〈p〉 f

)p

]= 4π

∂

∂p

(p2 〈p〉

)dp

for similar reasons than before.

ASP2018


RiaanSteenkamp

CR Continuity Equations VII

Thus∫Ωp·∇pF dΩ =

1

p2 dp

∫∇p·(pF ) d3p =

4π

p2

∂

∂p

(〈p〉 p2f

)Therefore

4πdf

dt= 4π

∂f

∂t+ 4π∇·(〈v〉 f ) +

4π

p2

∂

∂p

(〈p〉 p2f

)or by using Up = 4πp2f we have that

dUp

dt=∂Up

∂t+∇·Sp +

∂

∂p(〈p〉Up)

with Sp = 〈v〉Up the differential current density.

Normally dUp/dt = 0 and the above is basically acontinuity equation

ASP2018


RiaanSteenkamp

CR Continuity Equations VIII

If there exist a source or drain that adds or removesparticles to or from the system, one can write

∂Up

∂t+∇·Sp +

∂

∂p(〈p〉Up) = Qp

with Qp a differential source/drain term.

The abovementioned cosmic-ray continuity equation isoften called the “transport equation”

ASP2018


RiaanSteenkamp

Current Density in the Solar Wind I

Consider gas of charged particles with distribution functionF (r ,p, t) that satisfies the Boltzmann equation

∂F

∂t+ v ·∇F + a·∇pF =

(∂F

∂t

)c︸︷︷︸

collision term

.

with v & a particle velocity and acceleration, respectively. Ifwe set 〈v〉 = w we can write the convective derivative

dw

dt=∂w

∂t+ (w ·∇)w = 〈a〉 − 1

ρ

(∇·P

)+

1

ρδQ

with δQ =∫m(v −w)

(∂F∂t

)cd3v , ρ the mass density of gas

andP the pressure tensor of the gas

ASP2018


RiaanSteenkamp

Current Density in the Solar Wind II

Assumptions:

w measured w.r.t. irregular component of B-field.

Temporal changes & radients in w so small that dw/dtcan be neglected w.r.t. 〈a〉 = q

m (E + w×B)

Viscous effects of gas negligable ∴∇·P =∇P

Collision model describes particle scattering by B-fieldkinks with frequency ν = 1/τ , τ the collision time, and(∂F/∂t)c = −ν[F − F∗] where F∗ the equilibriumdistribution function. Now δQ/ρ can simply become −νw

With these the convective derivative for w can beapproximated as

0 ≈ q

m(E + w×B)− 1

ρ∇P − νw

ASP2018


RiaanSteenkamp

Current Density in the Solar Wind III

Solar wind a highly ionised plasma and thus in solar windreference frame there cannot be an E -field, and number densityin interval (p, p + dp) is the differential number densityUp:Thus P = UpkT , ρ = mUp and Sp = wUp, the differentialcurrent density of particles.Now

0 =q

mν(w×B)Up −

kT

mν∇Up −wUp

=q

mν(Sp×B)− kT

mν∇Up − Sp

Set the cyclotron frequency ω = qB/m and diffusion coeficientκ = kT/mν Now

Sp = −κ∇Up + ωτSp×eB

ASP2018


RiaanSteenkamp

Current Density in the Solar Wind IV

Or, if B = B ez

(Sp)x = −κ∂Up

∂x+ ωτ (Sp)y

(Sp)y = −κ∂Up

∂y+ ωτ (Sp)x

(Sp)z = −κ∂Up

∂z

with a little linear algebra we can write

(Sp)x = −(

1

1 + ω2τ2

)(κ∂Up

∂x+ ωτ

∂Up

∂y

)(Sp)y = −

(1

1 + ω2τ2

)(κ∂Up

∂y− ωτ ∂Up

∂x

)ASP2018


RiaanSteenkamp

Current Density in the Solar Wind V

Or, in vector notation

Sp = −κ (∇Up)‖ −κ

1 + ω2τ2(∇Up)⊥ −

ωτκ

1 + ω2τ2∇Up×eB

= −κ‖ (∇Up)‖ − κ⊥ (∇Up)⊥ − κ>∇Up×eB

= −K·∇Up

where

K =

κ⊥ κ> 0−κ> κ⊥ 0

0 0 κ‖

with κ‖ = κ, κ⊥ = κ‖/(1 + ω2τ2) & κ> = ωτκ⊥Please note that ‖ and ⊥ refers to the B-field direction!

ASP2018


RiaanSteenkamp

Current Density from Stationary Frame

To use Galilean transformation S ′p = Sp + VUp to transformfrom solar wind frame to stationary frame would be WRONG!!!CORRECT:

S ′p = Sp + C ~VUp

with the Compton-Getting factor

C = 1− 1

3Up

∂

∂p(pUp)

Note:

The C-G effect: an apparent anisotropy in particle spectradue to relative motion between observer and source.

(similar to Doppler shift in photon spectra)

Thus: Sp = CVUp −K·∇Up

ASP2018


RiaanSteenkamp

The Transport Equation I

Remember CR Continuity Equation

∂Up

∂t+∇·Sp +

∂

∂p(〈p〉Up) = Q(r , p, t)

In modulation studies Q = 0. Consider, 〈p〉Up, as a currentdensity in momentum space due to average rate of momentumchange:

〈p〉p

=1

3V ·G

with G = 1Up∇Up a density gradient in the differential number

density. Now

∂Up

∂t+∇·

(CVUp −

K·∇Up

)+

∂

∂p

(1

3V ·G pUp

)= 0

ASP2018


RiaanSteenkamp

The Transport Equation II

With a little algebra this becomes

∂Up

∂t+∇·

(VUp −

K·∇Up

)− 1

3(∇·V )

∂

∂p(pUp) = 0,

known as the Transport Equation, and it is very general.Note:

The diagonal elements ofK, i.e. κ‖ and κ⊥, describes

diffusion ‖ and ⊥ to B-field.

The off-diagonal elements ofK, i.e. κ>, describes particle

drifts.

There exists a drift current density SDp = κ>eB

×∇Up dueto density gradient.

ALL other drifts are implicit in PTPE: E×B-, curvature &gradient-, and current sheet drift.

ASP2018


RiaanSteenkamp

The Parker Transport Equation I

One can rewrite it i.t.o. the omni-directional distributionfunction f =

Up

4πp2 :

∂f

∂t=∇·

(K

(S)

·∇f

)(diffusion)

+∇·(K

(A)

·∇f

)(guiding centre drift)

− V ·∇f (convection)

+1

3(∇·V )

∂f

∂ln p(energy/momentum change)

+ Qf (r , p, t) (souce/sink term)

ASP2018


RiaanSteenkamp

The Parker Transport Equation II

Applied to Parker Spiral Field one can rewrite it in heliosphericspherical polars resulting in a parabolic PDE in time, particlerigidity, as well as three spatial coordinates.To find the specific terms one can evaluate gradients anddivergences in heliospheric polars and rotate the diffusiontensor into spherical polars by using standard rotation matricesw.r.t. the garden hose angle,

ψ = tan−1

(Ωr sin θ

V

).

Huge challenge:Figuring out

κα = (κα)0

(Be

B

)aα

β

(P

P0

)bα

, α = ‖,⊥

ASP2018


RiaanSteenkamp

The Parker Transport Equation III

The full equation:

∂f

∂t=κrr

∂2f

∂r2+κθθ

r2

∂2f

∂θ2+

κφφ

r2 sin2 θ

∂2f

∂φ2

+1

r(κrθ + κθr )

∂2f

∂θ∂r+

1

r sin θ(κrφ + κφr )

∂2f

∂φ∂r+

1

r2 sin θ(κθφ + κφθ)

∂2f

∂φ∂θ

+

[1

r2

∂

∂r(r2κrr ) +

1

r sin θ

∂

∂θ(sin θκθr ) +

1

r sin θ

∂κφr

∂φ− Vr

]∂f

∂r

+

[1

r2

∂

∂r(r2κrθ) +

1

r2 sin θ

∂

∂θ(sin θκθθ) +

1

r2 sin θ

∂κφθ

∂φ−

Vθ

r

]∂f

∂θ

+

[1

r2 sin θ

∂

∂r(r2κrφ) +

1

r2 sin θ

∂

∂θ(sin θκθφ) +

1

r2 sin θ

∂κφφ

∂φ−

Vφ

r sin θ

]∂f

∂φ

+1

3

[1

r2

∂

∂r(r2Vr ) +

1

r sin θ

∂

∂θ(sin θVθ) +

1

r sin θ

∂Vφ

∂φ

]∂f

∂ln P

+ Qf (r, θ, φ, P, t)

ASP2018


RiaanSteenkamp

The Parker Transport Equation IV

Assume azimuthal symmetry and a radial solar wind:

∂f

∂t=κrr

∂2f

∂r2+κθθ

r2

∂2f

∂θ2

+

[1

r2

∂

∂r(r2κrr ) +

1

r sin θ

∂

∂θ(sin θκθr ) − V

]∂f

∂r

+

[1

r2

∂

∂r(r2κrθ) +

1

r2 sin θ

∂

∂θ(sin θκθθ)

]∂f

∂θ

+1

3r2

∂

∂r(r2V )

∂f

∂ln P+ Qf (r, θ, P, t)

ASP2018


RiaanSteenkamp

The Parker Transport Equation V

Assume azimuthal symmetry, radial solar wind and particledrifts

∂f

∂t=κrr

∂2f

∂r2+κθθ

r2

∂2f

∂θ2

+

[1

r2

∂

∂r(r2κrr ) +

1

r sin θ

∂

∂θ(sin θκθr ) − vdr − V

]∂f

∂r

+

[1

r2

∂

∂r(r2κrθ) +

1

r2 sin θ

∂

∂θ(sin θκθθ) −

vdθ

r

]∂f

∂θ

+1

3r2

∂

∂r(r2V )

∂f

∂ln P+ Qf (r, θ, P, t)

where

vdr = −1

r sin θ

∂

∂θ(sin θκ> sinψ) = −

1

r sin θ

∂

∂θ(sin θκθr )

vdθ = +1

r

∂

∂r(rκ> sinψ) = −

1

r

∂

∂r(rκrθ)

ASP2018


RiaanSteenkamp

Diffusive Schock Acceleration I

Charged particles can undergo first-order Fermi (shock)acceleration at an astrophysical shock such as the solarwind termination shock

Shock can be entered as either a boundary condition or asa matching condition to match upstream and downstreammedia:

Distribution function must stay continuous: f − = f +

Flux that diverges from shock must have source on theshock:

S+ − S− = limε→0

∫ rs−ε

rs−εQ dr

ASP2018


RiaanSteenkamp

Diffusive Schock Acceleration II

Latter matching condition can in 2D be written as the PDE

(∂f

∂r

)−=κ+rr

κ−rr

(∂f

∂r

)+

−V− − V+

3κ−rr

∂f

∂ln P−κ−rθ

− κ+rθ

rsκ−rr

∂f

∂θ+

Q∗

κ−rr

Can solve over 3 regions: before shock (upstream), on theshock, and beyond the shock (downstream)

ASP2018


RiaanSteenkamp

Solutions of the The Parker Transport Equation I

Complete analytical solutions impossible.

Two analytical approximations:

Convection-Diffusion approximation:

Vf − κdfdr

= 0.

Force Field approximation: CVf − κ dfdr = 0 with the

Compton-Getting factor C = 1− 13Up

∂∂p (pUp) = − 1

3d ln fd ln p .

This now becomes

Vp

3

df

dp+ κ

df

dr= 0.

Numerical solutions more complete & self-consistent.

ASP2018


RiaanSteenkamp

Solutions of the The Parker Transport Equation II

Numerical solutions:

1971: Fisk — Steady-state, spherically symmetric (1D)solution:

a0∂2f

∂r2+ c0

∂f

∂r+ e0

∂f

∂lnP= 0

solved with Crank-Nicholson algorithm.

1973: Fisk —Steady-state 2D solution:

a0∂2f

∂r2+ b0

∂2f

∂θ2+ c0

∂f

∂r+ d0

∂f

∂θ+ e0

∂f

∂lnP= 0

solved with a Peaceman-Rachford Alternating DirectionImplicit (ADI) algorithm.

ASP2018


RiaanSteenkamp

Solutions of the The Parker Transport Equation III

Improvements to Fisk’s 1973 ADI:1975:

Moraal & Gleeson added more physics to solutionCecchini & Quenvy presented an independent ADI model

1979:

Moraal et al. added first drift calculationsJokipii & Kopriva presented independent drift model

1983:

Kota & Jokipii — Steady state 3D solutionPerko & Fisk — spherically symmetric time-dependentsolution

∂f

∂t= a0

∂2f

∂r2+ c0

∂f

∂r+ e0

∂f

∂lnP

solved with a modified Crank-Nicolson discretisation.

ASP2018


RiaanSteenkamp

Solutions of the The Parker Transport Equation IV

1986: Jokipii presented the first 2D time-dependent ADImodel model that can accelerate particles at anastrophysical shock — no applications were explored

1988: Potgieter & Moraal developed a steady state,spherically symmetric model that can also accelerateparticles.

1990: Le Roux expanded Fisk’s spherically symmetrictime-dependent modulation model to two dimensions.

1993: Steenkamp & Moraal presented an independent 2Dtime-dependent hybrid Locally One Dimensional (LOD)model that could accelerate particles at shocks and atregions of adiabatic compression.

ASP2018


RiaanSteenkamp

Anomalous Component of Cosmic Rays I

GCR = Galactic Cosmic RaysACR = Anomalous Cosmic RaysASP2018


RiaanSteenkamp

Anomalous Component of Cosmic Rays II

Hypothesis:

Interstellar thermal neutrals drift into heliosphere

Close to Sun they become singly ionised

Convected outward by solar wind towards solar windtermination shock

Accelerated to non-thermal energies at termination shock

Difuse back into heliospehere to be modulated intoobserved spectra

ASP2018


RiaanSteenkamp

Anomalous Component of Cosmic Rays III

ASP2018


RiaanSteenkamp

Anomalous Oxygen (1977/78 & 1987 solarminima)

ASP2018


RiaanSteenkamp

Anomalous Helium (1977/78 & 1987 solar minim)

ASP2018


RiaanSteenkamp

Closure

Thank you

ASP2018

The Parker Transport Equation€¦ · often called the \transport equation" ASP2018. The Parker...

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