Download 1 Energy dissipation in astrophysical plasmas

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Energy dissipation in astrophysical plasmas
The following presentation should give a summary of possible mechanisms, that can give
rise to temperatures in astrophysical plasmas. It will be classified in 3 chapters.
The first one will give a mathematical description for the contributions of dissipation and
energy transport.
The second chapter will treat various mechanisms that might dissipate energy.
The last chapter will mention possible ways of transporting energy in a plasma.
1.1
energy equation
The energy equation is a very useful tool to show, wether there are sources or sinks of
energy in an atmosphere. If we look at a plane parallel atmosphere we can write the
energy equation as
ρ·T ·
ds
= −L,
dt
(1.1)
where L is the energy loss function, and s the entropy per unit mass. For the case L = 0
the conservation of entropy is valid. If we express above equation for an ideal polytropic
gas, in terms of the internal energy, it reads
ρ·
dE p dρ
− ·
= −L.
dt
ρ dt
(1.2)
For an ideal polytropic gas the pressure reads as p = mρ kT . E is the internal energy, and
cv and cp the specific heat for constant volume and pressure respectively. If we take the
internal energy as E = cv T , and insert cp = cv + kmb and γ = ccvp for the polytropic gamma,
we receive the energy equation as
d
ργ
·
γ − 1 dt
Ã
p
ργ
!
= −L.
(1.3)
The energy loss function L contains positive terms for the conductive, radiative and
convective flux which are sources of energy. The negative terms can be all forms of
energy dissipation, on which we will focus in the following chapters. If we now look at
atmospheres containing plasma, we have to decide which sources and sinks we take into
considerations.
For looking at outer atmospheres, we can use the Plane Parrallel Approximation,
meaning that we disregard the curvature of the atmosphere for small surface elements.
We can now look at the timescales for changes in ρ, p and T , and decide wether or not
we take into account the various contributions of dissipation.
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2 ENERGY DISSIPATION
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2.1
Energy dissipation
Ohmic dissipation
For plasma moving at non relativistic speeds, in the presence of a magnetic field, we have
j = σ · (E + v × B).
(2.1)
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σ is the electric conductivity and can be approximated by σ = const T 2 . If we consider
a cube of the size l, the stored magnetic energy inside the cube is
W = L3 ·
B2
.
2µ
(2.2)
The inflow of electromagnetic energy into a volume, produces ohmic heat and kinetic
energy (work done by the lorentz force). Thus we have
−∇ · (E × H) =
j2
+ v · (j × B)
σ
(2.3)
By looking at a thin shell we can disregard the pressure gradient. Then the ratio between
ohmic energy and kinetic energy becomes 1. Now lets consider a typical atmosphere with
a thickness of 10 km and T = 104 K.
The typical timescale for the release of ohmic energy is τD = L2 µσ.
With above values we receive a time of 5 · 1011 sec.
Thats a very slow rate of energy dissipation, and thus we can see, that big flares can’t
be heated by ohmic dissipation only. On the other hand we see that the energy for small
regions is released much faster, since the size of the region has a quadratic effect on the
time. Thus the heating of spicules can be explained pretty well.
2.2
Acoustic wave heating
Acoustic waves are longitudinal waves. In a uniform medium, sound waves steepen,
because they propagate, according to a dispersion relation. Every part of the wave profile
moves with a different velocity. Thus the crests possess a higher velocity than the troughs.
The ambient sound speed can be calculated by
cs =
Ã
γkb T
mi
!1
2
.
(2.4)
If we denote the velocity amplitude as v1 , the crests and troughs have a relative velocity
of 2v1 .
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The trough is ahead λ2 and will be overtaken after t = 4vλ1 .
s
.
The distance the wave can travel at cs before shocking is d = λc
4v1
Thus short period waves evolve into shock waves over much smaller distances than long
period waves.
For a vertically stratified atmosphere, the distance for shock formation is much smaller
because the wave amplitude increases rapidly with altitude. If we take the kinetic energy
2
constant and notice, that its proportional to ρv21 , we see that changes in ρ can only be
balanced by changes in v1 . In an isothermal − atmosphere the density develops as
z
z
ρz = e− Λ , thus the velocity amplitude has to change like vz = e 2Λ .(Λ is the scale height)
The distance for a shock to form in a vertically stratified medium is now
τ cs 2
d = 2Λ · ln · 1 +
.
2(γ + 1)Λv1
Ã
!
(2.5)
The waves can now dissipate their energy through viscous, thermal or radiative losses.
The flux of energy transmitted by the shocks can be seen as the work done by the pressure.
Fz = ν · (p2 − p1 ) · v2 ·
t0
,
τ
(2.6)
where subscripts 2 and 1 denote the rear and front shock, and t0 is depending on the
shock profile.
Thus we can get the rate of Energy–dissipation
dF
νγ(γ + 1)p1 η̃ 3
=−
,
dz
12
(2.7)
1
is the fractional compression.
where η̃ = ρ2ρ−ρ
1
In the photosphere of the sun we have a source for acoustic waves, due to granulation.
These acoustic waves were proposed to heat the corona. But it turned out that the waves
cant reach the corona, since they are reflected at the transition zone. Thus the acoustic
waves are responsible for the vast temperature increase in the transition zone, but not for
the high temperatures in the corona.
2.3
Dissipation of magnetic fields
For the description of a plasma we need a combination of the equations of motion,
maxwells equations and the continuity equation.
We can make the following approximations :
The velocity of the particles vmat ¿ c
The field configuration doesnt change rapidly with time.
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The phase-velocity vph ¿ c
E
The plasma of consideration have good conductivity B
¿1
and Ohm‘s law j = σE is valid in the resting frame of reference of the plasma.
With these approximation we get the Ideal − MHD , which is a linear approximation.
4π
· j,
c
1 ∂B
,
∇×E = − ·
c ∂t
∇ · B = 0,
1
j = σ · (E + v × B).
c
∇×B =
(2.8)
(2.9)
(2.10)
(2.11)
If we now assume some velocity field, and look at the changes of the EM-field , we have
MH–kinematics.
We can now look at a moving plasma with finite σ and solve the system of equations. We
can reduce the system to following equation
c2
∂B
− ∇ × (v × B) =
· 4B,
∂t
4πσ
(2.12)
with values for vr,t and Bt0 we can calculate Bt . For v = 0 and B = (Bx(y,t) , 0, 0) we get
equation
c2 ∂ 2 Bx
∂Bx
=
·
.
∂t
4πσ ∂y 2
(2.13)
This equation is of the same type as the equation for thermal conductivity. There is
an analogy between lines of constant temperature and fieldlines. Eventhough the flux
through the y–z plane is constant, the magnetic energy of the system decreases. Because
of the finite σ, the magnetic energy is converted into heat.
For the timescale of the dissipation for magnetic fields we get
τ=
4πσ
· L2 .
2
c
(2.14)
Since the timescale is depending on the square of the size, we see that small field configurations dissipate fast, big ones slow.
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2.4
Heating by MHD-Waves
Additional to the equations 2.8 to 2.11 , we have to take into account the dynamic
interaction between fields and matter.
At first we need the Equation–of–motion for matter.
ρ·
∂v
1
+ ρ · (v · ∇)v = −∇P + · j × B + ρ g + ρν4v.
∂t
c
(2.15)
The second equation we need to add is the Equation–of–continuity which reads
∂ρ
+ ∇ · (ρv) = 0.
∂t
(2.16)
According to the nature of changes in the plasma we also have to define the Equation–
of–state, to describe adiabatic or polytropic changes.
⇒ Together with equations 1 to 4 we have linear homogeneous system of partial differential
equations. But since space and time coordinates do not appear explicitly in the equations,
we can make an exponential ansatz (B0 is homogenous in x-direction; σ → ∞; v0 = 0)
2.5
MHD-compressional-waves
Ansatz:
v1 =(0, v1y(y,t) , 0); v1y = c · exp (−iωt ± ily)
∂
∂x
= 0;
∂
∂z
= 0;
∂
∂y
= ±il;
∂
∂t
= −iω
Since v1y is only dependent on y, we have got a longitudinal wave. When such waves
propagate, we have a periodic compression of plasma and thus we need a relation for
adiabatic-changes
P1 = γ ·
P0
· ρ1 = (vs )2 ρ1 ,
ρ0
(2.17)
which yields the adiabatic–soundspeed
P0
vs = γ
ρ
·
¸1
2
.
(2.18)
Solving the system of equations yields a relationship for the phasevelocity
u2 =
ω2
= vs2 + va2 .
l2
(2.19)
In case of a small magnetic field the phase velocity converges to the adiabatic sound speed,
in case of a strong magnetic field, it converges to the Alfvén velocity.
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2.6
Alfvén-waves
Ansatz :
v1 = (0, v1y(x,t) , 0) ; v1y = c · exp (−iωt ± ilx)
and the magnetic field only in x direction
∂
∂z
=0;
∂
∂y
=0;
∂
∂t
= −iω
Since v1y is depending on x and t, our solution is a transversal wave.
With above ansatz we get the solution
4πω 2 ρ0
,
Bo2
(2.20)
ω
B0
=
1 .
l
(4πρ0 ) 2
(2.21)
l2 =
with the Alfvén–velocity
va =
When we look at the formula for the Alfvén–velovity we recognize that its only depending
on B0 and ρ0 . Thus we can identify it as a typical variable of state for a plasma configuration.If we have distortions, they propagate along the field lines.
For strong magnetic fields, va becomes big, for high densities it becomes small.
In the case where the density converges to 0 and va converges to c, Alfvén waves develope
into electromagnetic waves.
For both kinds of waves mentioned in this chapter we can see that the phase velocity is
independent of the frequency, meaning that the waves can propagate without dispersion!!!
If we make an ansatz in an arbitrary angle to the magnetic field, we get three solutions
namely Alfvén, MHD–compression and soundwaves. Thus in most cases we have a
combination of those wave types.
2.7
Magnetoacoustic–waves
Are a combination of MHD and acoustic waves.we distinguish between fast and slow mode
waves. Both kinds of waves form shocks and dissipate like sound waves, with ohmic dissipation as an additional energy source.
Fast–waves can transport energy in all directions. In regions where va À vs they propagate with va in all directions.
Slow–waves transport energy only in directions that are close to the magnetic field.
Alfvén-waves dissipate much less energy than magnetoacoustic–waves. But there is however a nonlinear interaction between Alfvén–waves. For a weak magnetic field andva < vs ,
two waves which propagate in opposite directions along a fieldline, can couple with each
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other. That results in the formation of a soundwave, which is able to dissipate its energy
faster.
In regions with a strong magnetic field an Alfvén–wave can split up into another one,
propagating in the opposite direction plus a sound–wave in the original direction. The
new formed Alfvén-wave has a lower frequency and can again split up...
That can lead to a cascade and contributes to a faster energy dissipation.
2.8
Reconnection in Current Sheets
A current sheet may be defined as a non propagating boundary between two plasmas. It
is a tangential discontinuity without any flow.
The total pressure on both sides of the sheet has to be continous
p2 +
B1 2
B2 2
= p1 +
.
2µ
2µ
(2.22)
The magnetic strength is the same on both sides, but with opposite directions.
At the center of the sheet the plasma pressure is enhanced by
p0 =
Bz1 2
Bz2 2
=
.
2µ
2µ
(2.23)
The magnetic field vanishes completely in the center of the field. Thus a current sheet
can be regarded as a discontinuity that separates 2 regions where the equations of ideal
MHD hold.
? In the absence of flow the current sheet will diffuse away at the speed ηl
? If we have a flow of plasma and magnetic flux towards the sheet with vi , we can
distinguish between 3 cases
η
vi >
expansion
(2.24)
l
η
steady
(2.25)
vi =
l
η
contraction
(2.26)
vi <
l
Now the enhanced pressure in the center of the sheet expels material at the ends of the
sheet.
Thus magnetic flux is ejected with the plasma, and one effect on the sheet is, to reconnect
its fieldlines. If the inflow speed is sub–Alfvénic, the outflow field strength is smaller than
the inflow.
B0 =
vi
· Bi .
va
(2.27)
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Figure 1: current sheets 1
Thus magnetic energy is converted into heat and flow energy.
This mechanism is very important for the heating of flares.
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3.1
Energy transport
Conductive energy transport
In high temperature regions, particles have a high velocity and kinetic energy. They may
travel into cooler regions, like particles of cooler regions may travel into hotter ones.This
process can be seen as a diffusion process. Therefore we have a net energy transport into
cooler regions.
The energy transport itself happens through collisions of particles. Conduction is depending on the following two points.
3.1.1 Density
If the density is low, the particles can travel further, which makes conduction more effective. On the other hand there are less particles that collide. These two effects can cancel
each other, and make conduction independent of density (which is the case in the corona
or transition zone)
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Figure 2: current sheets 2
3.1.2 Collisional crossection
Is depending on the particle species and velocity. For high velocities the crossection is
decreasing, which leads to increasing mean free path length.
In the presence of magnetic fields, it makes sense to split up the heat flux into into a component parallel and perpendicular to the magnetic field.Conduction along the magnetic
field is primarily governed by electrons, across the fields primarily by ions.
3.2
Energy transport by radiation
Figure 3: radiation
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Consider a surface area of size dσ, and radiation passing through it in a cone dω. The
amount of energy going through that area is
Eλ = Iλ cos Θdωdσdλ
(3.1)
Figure 4: radiation 2
If we imagine a volume element of plasma where radiation passes through, we have absorption and emission. Emission gives rise to the energy by
dEλ = ελ dωdσdλds.
(3.2)
The Absorption reduces the energy by
dEλ = −κλ Eλ ds = −κλ Iλ dωdσdλds.
(3.3)
Out of the two equations for emission and absorption we get the Radiative–transfer–
equation which reads
dIλ
ελ
= −Iλ + .
κλ ds
κλ
(3.4)
The term κελλ is called the Source–function and plays a very important roll in stellar
astrophysics. It can be seen as the ration between reemission and absorption in an atmosphere. The expression κλ ds is also called the optical depth. It is necessary for the sun,
to define an optical depth that we define as surface of the sun.
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3.3
Convective energy transport
In an adiabatic process we have an Adiabatic–gamma
γ=
cp
.
cv
(3.5)
For the pressure in the plasma we have the relation
P = const · ργ =
ρ
· kT.
µmh
(3.6)
Out of above relations for the pressure we can derive an Adiabatic–temperature–
gradient describing an adiabatic changes in the plasma.
∇ad =
1
T
1
P
·
·
dT
dr
dP
dr
=
γ−1
.
γ
(3.7)
We can now consider a bubble, which is moved by a force the distance ∆r, undergoing
an adiabatic change, keeping it in pressure equilibrium with its surrounding. Thus the
temperature of the bubble will change according to the adiabatic temperature gradient.
T2Bubble
dT
= T1Bubble +
dr
·
¸
(3.8)
·∆r.
adiabatic
The temperature of the surrounding plasma will change according to
T2Surrounding
dT
= T1Surroundingbubble +
dr
·
¸
·∆r.
(3.9)
external
For a bubble to start moving on its own, we have to have a marginally lower density, and
a marginally higher temperature.
If we insert ρ1B = ρ1 − δρ and T1B = T1 + δT into the gas law, using the constrain that
ρ1B ≤ ρ1 , we receive the Criterion–for–Convection.
·
dT
dr
¸
adiabatic
>
·
dT
dr
¸
(3.10)
external
In atmospheres this might be the case in areas where the ionisation degree changes rapidly.
increases. If we consider a
From γ = ccvp = 1 + Ncvk follows that γ will decrease if cv = dU
dT
non ideal gas, the inner energy might also include terms for excitation and ionisation.
In areas where the ionisation degree changes rapidly we have dU
> 32 · N k which leads to
dT
γ < 53 and thus ∇ad < 0, 4. That might lead to convection.