Alfven waves are the most important waves propagating in the solar atmosphere, as well as the Earth’s magnetosphere (underpinning the coupling between it and the ionosphere). They are important in that they efficiently carry energy and momentum along the magnetic field.

One way to get a handle (of sorts) on
Alfven waves is to look at the analogy with mechanical waves – say propagating
along in a single direction x for a
string put under tension T.

Consider the reference frame or coordinate
system for a transverse wave on a string:

Where we have in terms of the
string tension **T**:

**T _{u} = T sin **

**q**

As usual per these approaches, assume q is* very small*, in which
limit, *sin **q** **»** tan **q*

Then:

**T sin q
= T tan q = T (¶ u/ ¶****x)**

Taking the force difference:

**[T _{u} ] _{x + dx} - [T _{u}
] _{x} = ¶ / ¶x (T (¶ u/ ¶**

**x) dx**

To ensure *no net horizontal forces*
due to tension, we limit the
situation to small slopes. Also, neglect the possibility of a vertical force
per unit length, so:

**r dx (¶ ^{2 }u/ ¶ t^{2}) = ¶ / ¶x (T (¶ u/ ¶x) dx**

Which simplifies to:

**r (¶ ^{2 }u/ ¶ t^{2}) = T (¶ ^{2}
u/ ¶x^{2} )**

Now, transform to the standard wave equation in 1-dimension:

**¶ ^{2} u/ ¶x^{2} - 1/ c^{2}
(¶ ^{2 }u/ ¶ t^{2}) = 0**

From this we can solve for the velocity c:

**c = (T/ r****
) ^{½}**

Say x marks the direction of
propagation in the above coordinate system, and y is the direction of
transverse (wave) displacement. Then the vertical force component is:

**F _{y} = - T(¶ y/ ¶**

**x)**

where T
is the tension. Thus, just as the restoring force for a mechanical wave is the
string tension T, the restoring force for an Alfven wave is the magnetic
tension. This magnetic version of “tension” accelerates the plasma and is
opposed by the inertia of the ions (mainly from proton masses m _{p})

Now, the wave speed on a string is
related to u (mass per unit length), and T such that:

**v = (T/ u) ^{1/2}**

And as we can see, increasing the
string tension T ** _{u}** increases the wave speed in an analogous way to
what magnetic tension does for the Alfven wave. The magnetic tension analog can
be expressed (as we shall see) as:

**T _{M} = B^{2} / m**

_{o}

where B is the magnetic induction
and m_{o} is the magnetic permeability for free space ( 4π x
10 ** ^{-7}**
H/m)

In what
follows we assume a uniform plasma in equilibrium, which will then be subjected
to velocity disturbance or perturbation that affects all other key quantities.
The treatment is kept as simple as possible (considering the complexity of the
subject matter!) , and we don’t veer out of the linear domain. Nevertheless it
should be stated at the outset that some details are omitted, or left as work
for yourself with hints provided. In this way you will better understand and
appreciate the genesis of Alfven waves.

Examining
the origin of these waves always starts with setting out the basic equations
for what we call “ideal MHD”:

1) ¶
**r** / ¶ t = - DIV (**r**
v)

2) ¶ (**r** v)/ ¶ t =

- DIV (**r**
vv) – grad p + 1/ m _{o} [(**Curl B**) **X B**]

3) ¶
**B** / ¶t = **Curl**
(**v X B**)

4) ¶p/ ¶ t = - v. grad p – g p DIV v

Where v is the fluid velocity, p the
pressure, **B**
the magnetic induction, and g = - d ln p/ d ln V where V denotes
volume. Next we introduce small perturbed quantities (e.g.
imagine introducing a small perturbation into the plasma velocity such that v_{o}
< < v _{l }, which will also subject the mass density, fluid
pressure and magnetic field to perturbation), such that:

**r ** = **r**_{o
}+ **r**
_{l}

v = v _{l } , B
= B _{o} + B _{l } , p =
p _{o} + p _{l}

Now, substitute these back
into the original ideal MHD equations to obtain:

5) ¶ **r**** **_{l} / ¶
t = - **r **_{o} **DIV** v _{l}

6) **r**_{o } (¶v _{l}/ ¶ t) = - grad p _{l} +
1/ m _{o } [(**Curl
B **** _{l}**)

**X B**

_{o}_{ }]

7) ¶ **B **
_{l} / ¶t
= **Curl** (v _{l}**
****X B **_{o })

8) ¶ p** **
_{l} / ¶t = -
g p _{o} DIV v _{l}

Now, divide through the 2nd
equation above by the mass density **r**:

¶v _{l} / ¶t
= -(c _{s}** ^{2}**)
grad p

_{l}/

**r**- 1/ m

_{o }

**r**[

**B**

_{o}x Curl B**]**

_{1}

where ‘**c _{s}**’ is the
sound speed. Using this result and the last two equations of the perturbed set,
we apply Fourier transforms to obtain:

w^{2 } v _{l} – c _{s} ** ^{2 }** (kv

_{x })k* +

**B **** _{o }**/ m

_{o }

**r**

_{o}[

**k X k***X (

**v**

_{l }X

**B**

**)] = 0**

_{o}Where w denotes the plasma
frequency, k is the wave number vector (**k***
the wave vector orientation) and the other quantities are as before.

It is
of interest here to obtain the x, y-components of velocity associated with the
wave, by use of Fourier transforms. The
procedure is straightforward and I show it in what follows.

The
x-component is:

w^{2 } v _{x} – c _{s} ** ^{2 }** k

^{2 }**v**

^{ }_{x }+ B

_{z}k

**/ m**

^{2 }_{o }

**r**

_{o}

**[v**

_{z}B

_{x}– v

_{x}B

_{z}] = 0

The y-component is decoupled from
the others (e.g. x, z) and can be written:

w^{2 } **v** _{y } -
B _{x} ** ^{2 }**
k

^{2 }**v**

_{y }/ m

_{o }

**r**

_{o}= 0

Or simply:

w^{2 } =
[B _{x} ** ^{2
}**/ m

_{o}

**r**

_{o}] k

^{2}where the quantity in brackets is
the **Alfven velocity** or alternatively written:

**v** _{A } = [ w/ k] = B _{x } / [m _{o} **r **_{o}] ^{1/ 2}

Or :

**v ** _{A } = B_{o} / [m _{o} **r**** **_{o}]
^{1/ 2}

(Since **B**_{o} is in the x –direction)

For completeness, we should be
able to show the z-component equation is:

w^{2 } v _{z } - B _{x } k ** ^{2
}**/ m

_{o}

**r**

_{o}[v

_{z}B

_{x}- v

_{x}B

_{z}] = 0

I refer again to the basic wave equation one can
obtain by getting the second derivative of:

¶v _{l} / ¶t
= - (c _{s} ** ^{2
}**) grad p

_{l}/

**r**- 1/ m

_{o }

**r**

**[**

**B**

_{o}**X Curl B**

**]**

_{1}One can then find the solution in terms of plane waves by assuming:

**v** _{1} = **v ** _{1 }[exp ikx – iwt]

For
which taking the second derivative, of v _{1} with
respect to t yields the original equation in w found earlier.

For completeness, I note what
happens when you solve the preceding (simultaneous) equations in x, and z.

w ** ^{4}** +
w

^{2 }[- c

_{s}

**k**

^{2 }**- B**

^{2 }_{x}

**k**

^{2 }**/ m**

^{2 }_{o}

**r**

_{o }] +

c _{s} ** ^{2
}** k

^{4 }**[B**

^{ }_{x}

**/ m**

^{2 }_{o}

**r**

_{o }] = 0

Or:

w ** ^{4}** -
w

^{2 }(c

_{s}

**+ v**

^{2 }_{A }

**)k**

^{2 }**+ c**

^{2 }_{s}

**v**

^{2}_{A }

**cos**

^{2 }

^{ }**(Θ) k**

^{2}**= 0**

^{4 }And finally,

w^{2 } = ½[(
c _{s }** ^{2}** v

_{A }

**k**

^{2 }

^{2}__+__

[(c _{s} ** ^{2}** v

_{A }

**k**

^{2 }**– 4 c**

^{4}_{s}

**v**

^{2}_{A }

**cos**

^{2 }**(Θ) k**

^{2 }**]**

^{4}

^{1/2 }Careful variation of Θ for
the above equations leads to special conditions which can also identify the
“fast” and “slow” megnetosonic waves

The conditions and associated
results are expressed in summary form:

For Θ = 0, w^{2 } = { v _{A }^{2 }k ^{2 }

{ c _{s }^{2} ** ^{ }**k

^{2}For Θ = 90, w^{2 } = {(c _{s} ** ^{2 }**
+ v

_{A }

**)k**

^{2 }

^{2 }= {0

It is convenient to distinguish these waves of differing velocity using the diagram shown below:

Note in particular the orientation of the wave number vector
(k) as well as the lobes for the sound speed
c _{s}_{ } and
the Alfven velocity v _{A} - both in relation to (c _{s} ** ^{2 }** +
v

_{A }

**) and the equilibrium magnetic field intensity, B**

^{2 }_{o. }Note (c

_{s}

**+ v**

^{2 }_{A }

**) is called the “fast mode” magneotsonic wave velocity, while c**

^{2}_{s}denotes the “slow mode” velocity. If one plots the preceding using (c

_{s}

**+ v**

^{2 }_{A }

**) for the vertical axis and B**

^{2 }_{o}(e.g. x) for the horizontal, one will also get what is called “

*” shown below:*

**Friedrich’s diagram** The
__Friedrich diagram__ consists of :

1)A smaller “dumb bell” or
figure-8 shaped graph centered at the origin. This will be for what we call
“slow mode” waves

2)A double lobe enveloped *by a larger circle* and itself enveloping
a smaller double lobe. This will be for Alfven waves proper.

3) A circle shaped graph
surrounding both 1, 2 above. This will be for what we call the “fast MHD” mode.

The critical aspect to note here is that the
fast mode is the only MHD wave able to carry energy *perpendicular* to the magnetic field. This has important
ramifications for solar flares, as well as magnetospheric effects (such as the
aurora). Meanwhile, the phase velocity (w/k) of the slow mode wave
perpendicular to the magnetic field is always zero. In the limit where the
sound speed:

c _{s} ** ^{2 }** > >
v

_{A }

**, and the Alfven speed:**

^{2} v _{A
}** ^{2}** < < c

_{s}

**,**

^{2}The
slow wave disappears. (Which one can easily validate and confirm for the
equation in w^{2 })

Other properties, points to note:

-The velocity
perturbation v _{1} is orthogonal to B _{o}

-The wave is incompressible since **DIV** **v** _{1}
= ik**.v** _{1} = 0

-The magnetic field perturbation (**B** _{1}) is aligned with the
velocity perturbation. Since both are perpendicular to k and B _{o}

-The current density perturbation
(**J** _{1}) exists as a
current perturbation perpendicular to k and B _{o} e.g.

**J** _{1} **= k X B** _{o}

- When c _{s} ** ^{2}**
>> v

_{A }

**the fast mode wave becomes a compressional Alfven wave. This has a group velocity equal to its phase velocity w /k**

^{2}For fast mode waves with k ⊥ B _{o }we can present the diagrams below
showing the relations between the distinct field directions and applicable
velocity as well as wave number vector in relation to B-field perturbations.

1) For the mechanical wave diagram for transverse wave on a string sketch *the x, y-components* of the string
tension T. Show how the components and differences between them lead to the
result:

[T ** _{u}** ]

**- [T**

_{x + dx}**]**

_{u}**= ¶ / ¶x (T (¶ u/ ¶x) dx**

_{x}2) (a) A student sketches the diagram below to show the different modes of plasma waves which he argues exist in the Sun.

Redraw the diagram to improve its clarity and then explain the reasons for any changes made. Would the parameters shown really make sense for such waves propagating in the Sun? Why or why not? Give a quantitative explanation if possible.

(b)Compute the Alfven velocity in a layer of the Sun for which:

**r
**_{o} = 0. 2 g/ cm ** ^{3}** and B = 0.1 T.

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