**Chapter 5: Waves and characteristics**

## Overview

*•*

**Physics and accounting:**use example of sound waves to illustrate method of lin-earization and counting of variables and solutions; [ book: Sec. 5.1 ]

*•*

**MHD waves:**different representations and reductions of the linearized MHD equations, obtaining the three main waves, dispersion diagrams; [ book: Sec. 5.2 ]

*•*

**Phase and group diagrams:**propagation of plane waves and wave packets, asymp-totic properties of the three MHD waves; [ book: Sec. 5.3 ]

*•*

**Characteristics:**numerical method, classification of PDEs, application to MHD.

## Sound waves

*•*

Perturb the gas dynamic equations (**B = 0**

),
*∂ρ*

*∂t*

**+ ∇ · (ρv) = 0 ,**

**+ ∇ · (ρv) = 0 ,**

### (1)

*ρ*

**∂v**

**∂v**

*∂t*

**+ v · ∇v**

**+ v · ∇v**

*+ ∇p = 0 ,*

### (2)

*∂p*

*∂t*

**+ v · ∇p + γp∇ · v = 0 ,**

**+ v · ∇p + γp∇ · v = 0 ,**

### (3)

about infinite, homogeneous gas at rest,

**ρ(r, t) = ρ**

0 **ρ(r, t) = ρ**

*+ ρ*

1**(r, t)**

**(r, t)**

*(where |ρ*

1*| ρ*

0 *= const) ,*

**p(r, t) = p**

0 **p(r, t) = p**

*+ p*

1**(r, t)**

**(r, t)**

*(where |p*

1*| p*

0 *= const) ,*

### (4)

**v(r, t) =**

**v(r, t) =**

**v**

1**(r, t)**

**(r, t)**

**(since v**

0 *= 0) .*

*⇒*

*Linearised equations of gas dynamics:*

*∂ρ*

1
*∂t*

*+ ρ*

0**∇ · v**

1 **∇ · v**

*= 0 ,*

### (5)

*ρ*

0 **∂v**

1
**∂v**

*∂t*

*+ ∇p*

1 *= 0 ,*

### (6)

*∂p*

1
*∂t*

*+ γp*

0**∇ · v**

1 **∇ · v**

*= 0 .*

### (7)

## Wave equation

*•*

Equation for _{ρ}

_{ρ}

_{1}does not couple to the other equations: drop. Remaining equations give

*wave equation for sound waves:*

*∂*

2**v**

1
*∂t*

2 *− c*

2_{∇∇ · v}

1

_{∇∇ · v}*= 0 ,*

### (8)

where*c ≡*

*γp*

0*/ρ*

0 ### (9)

is *the velocity of sound* of the background medium.

*•*

Plane wave solutions
**v**

1**(r, t) =**

**(r, t) =**

**k**

### ˆ

**v**

**k**

*e*

**i(k·r−ωt)**### (10)

turn the wave equation (8) into an algebraic equation:

*( ω*

2**I − c**

2**I − c**

**kk ) · ˆv = 0 .**

**kk ) · ˆv = 0 .**

### (11)

*•*

For **k = k e**

**k = k e**

*, the solution is:*

_{z}*ω = ±k c ,*

*ˆv*

*x*

*= ˆv*

*y*

*= 0 ,*

*ˆv*

*z*

### arbitrary

*,*

### (12)

*⇒*

*Sound waves*propagating to the right (

_{+}

) and to the left (*−*

):
*compressible*(

**∇ · v = 0**

) and **∇ · v = 0**

*longitudinal*(

**v k**

) waves.
**v k**

## Counting

*•*

There are also other solutions:
*ω*

2 *= 0 ,*

*ˆv*

_{x}*, ˆv*

_{y}### arbitrary

_{,}

_{,}

_{ˆv}

_{ˆv}

_{z}_{= 0 ,}

_{= 0 ,}

### (13)

*⇒*

*incompressible transverse (*

**v**

_{1}

**⊥ k**

**⊥ k**

*) translations.*They do not represent interesting physics, but simply establish completeness of the velocity representation.

*•*

Problem: 1st order system (5)–(7) for _{ρ}

_{ρ}

_{1},

**v**

_{1},

_{p}

_{p}

_{1}has 5 degrees of freedom, whereas 2nd order system (8) for

**v**

_{1}appears to have 6 degrees of freedom (

_{∂}

2_{∂}

_{/∂t}

2 _{/∂t}

*→ −ω*

2).
However, the 2nd order system actually only has 4 degrees of freedom, since _{ω}

2 does
not double the number of translations (13). Spurious doubling of the eigenvalue _{ω}

_{ω = 0}

happened when we applied the operator _{ω = 0}

_{∂/∂t}

to Eq. (6) to eliminate _{∂/∂t}

_{p}

1.
_{p}

*•*

Hence, we *lost one degree of freedom*in the reduction to the wave equation in terms of

**v**

_{1}alone. This happened when we dropped Eq. (5) for

_{ρ}

_{ρ}

_{1}. Inserting

**v**

_{1}

_{= 0}

in the
original system gives the signature of this lost mode:
*ω ˆ*

*ρ = 0*

**⇒ ω = 0 , ˆρ arbitrary , but ˆv = 0 and ˆp = 0 .**

**⇒ ω = 0 , ˆρ arbitrary , but ˆv = 0 and ˆp = 0 .**

### (14)

*⇒*

*entropy wave: perturbation of the density and, hence, of the entropy*

_{S ≡ pρ}

_{S ≡ pρ}

*−γ*. Like the translations (13), this mode does not represent important physics but is needed to account for the degrees of freedom of the different representations.

## MHD waves

*•*

Similar analysis for MHD in terms of _{ρ}

, _{ρ}

**v**

, _{e}

_{e}

*≡*

*1*

_{γ−1}_{p/ρ}

, and _{p/ρ}

**B**

:
*∂ρ*

*∂t*

**+ ∇ · (ρv) = 0 ,**

**+ ∇ · (ρv) = 0 ,**

### (15)

*ρ*

**∂v**

**∂v**

*∂t*

**+ ρv · ∇v + (γ − 1)∇(ρe) + (∇B) · B − B · ∇B = 0 ,**

**+ ρv · ∇v + (γ − 1)∇(ρe) + (∇B) · B − B · ∇B = 0 ,**

### (16)

*∂e*

*∂t*

**+ v · ∇e + (γ − 1)e∇ · v = 0 ,**

**+ v · ∇e + (γ − 1)e∇ · v = 0 ,**

### (17)

**∂B**

**∂B**

*∂t*

**+ v · ∇B + B∇ · v − B · ∇v = 0 ,**

**+ v · ∇B + B∇ · v − B · ∇v = 0 ,**

**∇ · B = 0 ,**

**∇ · B = 0 ,**

### (18)

*•*

*Linearise about plasma at rest,*

**v**

_{0}

_{= 0 , ρ}

_{= 0 , ρ}

_{0}

_{, e}

_{, e}

_{0}

_{, B}

_{, B}_{0}

_{= const}

:
*∂ρ*

1
*∂t*

*+ ρ*

0**∇ · v**

1 **∇ · v**

*= 0 ,*

### (19)

*ρ*

0 **∂v**

1
**∂v**

*∂t*

*+ (γ − 1)(e*

0*∇ρ*

1 *+ ρ*

0*∇e*

1**) + (∇B**

1**) + (∇B**

**) · B**

0 **) · B**

**− B**

0 **− B**

**· ∇B**

1 **· ∇B**

*= 0 ,*

### (20)

*∂e*

1
*∂t*

*+ (γ − 1)e*

0**∇ · v**

1 **∇ · v**

*= 0 ,*

### (21)

**∂B**

1
**∂B**

*∂t*

**+ B**

0**∇ · v**

1 **∇ · v**

**− B**

0 **− B**

**· ∇v**

1 **· ∇v**

*= 0 ,*

**∇ · B**

1 **∇ · B**

*= 0 .*

### (22)

## Transformation

*•*

*Sound*and vectorial

*Alfv ´en speed*,

*c ≡*

*γp*

0
*ρ*

0 *,*

**b ≡**

**b ≡**

**B**

0
*√*

*ρ*

0 *,*

### (23)

and dimensionless variables,

### ˜

*ρ ≡*

*ρ*

1
*γ ρ*

0 *,*

**v ≡**

**v ≡**

### ˜

**v**

1
*c*

*,*

*˜e ≡*

*e*

1
*γ e*

0 *,*

### ˜

**B ≡**

**B ≡**

**B**

1
*c*

*√*

*ρ*

0 *,*

### (24)

*⇒*

*linearised MHD equations*with coefficients

_{c}

and _{c}

**b**

:
*γ*

*∂ ˜*

*ρ*

*∂t*

*+ c ∇ · ˜*

**v = 0 ,**

**v = 0 ,**

### (25)

*∂ ˜*

**v**

*∂t*

*+ c ∇˜*

*ρ + c ∇˜e + (∇ ˜*

**B) · b − b · ∇ ˜B = 0 ,**

**B) · b − b · ∇ ˜B = 0 ,**

### (26)

*γ*

*γ − 1*

*∂ ˜e*

*∂t*

*+ c ∇ · ˜*

**v = 0 ,**

**v = 0 ,**

### (27)

*∂ ˜*

**B**

*∂t*

**+ b ∇ · ˜**

**+ b ∇ · ˜**

**v − b · ∇˜v = 0 ,**

**v − b · ∇˜v = 0 ,**

**∇ · ˜B = 0 .**

**∇ · ˜B = 0 .**

### (28)

**
**

## Symmetry

*•*

*Plane wave solutions, with*

**b**

and **k**

arbitrary now:
### ˜

*ρ = ˜*

**ρ(r, t) = ˆ**

**ρ(r, t) = ˆ**

*ρ e*

**i(k·r−ωt)***, etc.*

### (29)

yields an algebraic system of eigenvalue equations:

**c k · ˆ**

**c k · ˆ**

**v**

### =

*γ ω ˆ*

*ρ ,*

**k c ˆρ**

**k c ˆρ**

**+ k c ˆe + (kb · − k · b) ˆ**

**+ k c ˆe + (kb · − k · b) ˆ**

**B =**

*ω ˆ*

**v ,**

**v ,**

### (30)

**c k · ˆ**

**c k · ˆ**

**v**

### =

_{γ−1}γ*ω ˆe ,*

**(bk · − b · k) ˆ**

**(bk · − b · k) ˆ**

**v**

### =

*ω ˆ*

**B ,**

**B ,**

**k · ˆB = 0 .**

**k · ˆB = 0 .**

*⇒*

**Symmetric eigenvalue problem!**(The equations for

_{ρ}

_{ρ}

_{ˆ}

, **v**

_{ˆ}

, _{ˆe}

, and _{ˆe}

**B**

### ˆ

appear to know about each other.) .*•*

The symmetry of the linearized system is closely related to an analogous property of
the original nonlinear equations: *the nonlinear ideal MHD equations are symmetric*

*hyperbolic partial differential equations.*

## Matrix eigenvalue problem

*•*

Choose **b = (0, 0, b) , k = (k**

**b = (0, 0, b) , k = (k**

_{⊥}_{, 0, k}

_{, 0, k}

_{}_{)}

:
### ⎛

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎝

### 0

*k*

*⊥*

*c*

### 0

*k*

*c*

### 0

### 0

### 0

### 0

*k*

*⊥*

*c*

### 0

### 0

### 0

*k*

*⊥*

*c −k*

*b*

### 0

*k*

*⊥*

*b*

### 0

### 0

### 0

### 0

### 0

### 0

*−k*

*b 0*

*k*

_{}*c*

### 0

### 0

### 0

*k*

_{}*c*

### 0

### 0

### 0

### 0

*k*

*⊥*

*c*

### 0

*k*

*c*

### 0

### 0

### 0

### 0

*0 −k*

_{}*b*

### 0

### 0

### 0

### 0

### 0

### 0

### 0

### 0

*−k*

*b 0*

### 0

### 0

### 0

### 0

### 0

*k*

*⊥*

*b*

### 0

### 0

### 0

### 0

### 0

### 0

### ⎞

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎠

### ⎛

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎝

### ˆ

*ρ*

*ˆv*

_{x}*ˆv*

*y*

*ˆv*

_{z}*ˆe*

### ˆ

*B*

*x*

### ˆ

*B*

_{y}### ˆ

*B*

*z*

### ⎞

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎠

*= ω*

### ⎛

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎜

### ⎝

*γ ˆ*

*ρ*

*ˆv*

_{x}*ˆv*

*y*

*ˆv*

_{z}*γ*

*γ−1*

*ˆe*

### ˆ

*B*

*x*

### ˆ

*B*

_{y}### ˆ

*B*

*z*

### ⎞

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎟

### ⎠

*. (31)*

*⇒*

**Another representation of the symmetry of linearized MHD equations.**

*•*

New features of MHD waves compared to sound: occurrence of Alfv ´en speed _{b}

and
anisotropy expressed by the two components _{b}

_{k}

_{k}

*and*

_{}_{k}

_{k}

*of the wave vector. We could compute the dispersion equation from the determinant and study the associated waves, but we prefer again to exploit the much simpler velocity representation.*

_{⊥}

**
**

## MHD wave equation

*•*

Ignoring the magnetic field constraint **k · ˆB = 0**

in the **k · ˆB = 0**

_{8 × 8}

eigenvalue problem (31)
would yield _{8 × 8}

*one spurious eigenvalue*

_{ω = 0}

. This may be seen by operating with the
projector _{ω = 0}

**k·**

onto Eq. (30)(d), which gives **k·**

_{ω k · ˆ}

_{ω k · ˆ}**B = 0**

.
*•*

Like in the gas dynamics problem, a *genuine but unimportant marginal entropy mode*is obtained for

_{ω = 0}

with _{ω = 0}

**v = 0**

_{ˆ}

, _{p = 0}

_{p = 0}

_{ˆ}

, and **B = 0**

### ˆ

:*ω = 0 ,*

*p = ˆe + ˆ*

### ˆ

*ρ = 0 ,*

*S = γˆe = −γ ˆ*

### ˆ

*ρ = 0 .*

### (32)

*•*

Both of these marginal modes are eliminated by exploiting *the velocity representation.*The perturbations

_{ρ}

_{ρ}

_{1},

_{e}

_{e}

_{1},

**B**

_{1}are expressed in terms of

**v**

_{1}by means of Eqs. (19), (21), and (22), and substituted into the momentum equation (20). This yields the

**MHD**

**wave equation for a homogeneous medium**:

*∂*

2**v**

1
*∂t*

2 *−*

**(b · ∇)**

2 **(b · ∇)**

**I + (b**

2 **I + (b**

*+ c*

2**) ∇ ∇ − b · ∇ (∇ b + b ∇)**

**) ∇ ∇ − b · ∇ (∇ b + b ∇)**

**· v**

1 **· v**

*= 0 .*

### (33)

The sound wave equation (8) is obtained for the special case**b = 0**

.
## MHD wave equation (cont’d)

*•*

Inserting plane wave solutions gives the required eigenvalue equation:
*ω*

2 **− (k · b)**

2**− (k · b)**

**I − (b**

2 **I − (b**

*+ c*

2**) k k + k · b (k b + b k)**

**) k k + k · b (k b + b k)**

**· ˆv = 0 ,**

**· ˆv = 0 ,**

### (34)

or, in components:### ⎛

### ⎜

### ⎜

### ⎝

*− k*

2
*⊥*

*(b*

2 *+ c*

2*) − k*

2*b*

2 ### 0

*−k*

*⊥*

*k*

*c*

2
### 0

*−k*

*2*

_{}*b*

2 ### 0

*− k*

*⊥*

*k*

*c*

2 ### 0

*−k*

*2*

_{}*c*

2
### ⎞

### ⎟

### ⎟

### ⎠

### ⎛

### ⎜

### ⎝

*ˆv*

*x*

*ˆv*

_{y}*ˆv*

*z*

### ⎞

### ⎟

*⎠ = −ω*

2
### ⎛

### ⎜

### ⎝

*ˆv*

*x*

*ˆv*

_{y}*ˆv*

*z*

### ⎞

### ⎟

*⎠ .*

### (35)

Hence, a

_{3 × 3}

symmetric matrix equation is obtained in terms of the variable _{3 × 3}

**v**

_{ˆ}

,
*with quadratic eigenvalue*

_{ω}

2, corresponding to the original _{ω}

_{6 × 6}

representation with
eigenvalue _{6 × 6}

_{ω}

(resulting from elimination of the two marginal modes).
_{ω}

*•*

Determinant yields the **dispersion equation:**

### det =

*ω*

*(ω*

2 *− k*

*2*

_{}*b*

2### )

*ω*

4 *− k*

2*(b*

2 *+ c*

2*) ω*

2 *+ k*

*2*

_{}*k*

2*b*

2*c*

2
### = 0

### (36)

## Roots

**1) Entropy waves:**

*ω = ω*

*E*

*≡ 0 ,*

### (37)

### ˆ

**v = ˆB = 0 , ˆp = 0 , but ˆs = 0 .**

**v = ˆB = 0 , ˆp = 0 , but ˆs = 0 .**

### (38)

*⇒*

just perturbation of thermodynamic variables.
**2) Alfv ´en waves:**

*ω*

2 *= ω*

*2*

_{A}*≡ k*

*2*

_{}*b*

2 *→ ω = ±ω*

_{A}*,*

### (39)

*ˆv*

*x*

*= ˆv*

*z*

### = ˆ

*B*

*x*

### = ˆ

*B*

*z*

*= ˆs = ˆ*

*p = 0 ,*

*B*

### ˆ

*y*

*= −ˆv*

*y*

*= 0 .*

### (40)

*⇒*

transverse **v**

### ˆ

and**B**

### ˆ

so that field lines follow the flow.**3) Fast (**

_{+}

**) and Slow (**

*−*

**) magnetoacoustic waves:**

*ω*

2 *= ω*

*2*

_{s,f}*≡*

1_{2}

*k*

2*(b*

2 *+ c*

2### )

*1 ±*

*1 −*

*4k*

2
*b*

2*c*

2
*k*

2*(b*

2 *+ c*

2### )

2*→ ω =*

*±ω*

*s*

*±ω*

*f*

### (41)

*ˆv*

*y*

### = ˆ

*B*

*y*

*= ˆs = 0 ,*

*but ˆv*

*x*

*, ˆv*

*z*

*, ˆ*

*p , ˆ*

*B*

*x*

*, ˆ*

*B*

*z*

*= 0 ,*

### (42)

## Eigenfunctions

**(x)**

**k**(y) (z) ϑ

**v**

_{A}

**B**

_{A}

**B**

_{0}(z) (x)

**k**ϑ

**v**

_{s}

**B**

_{s,f}

**B**

_{0}

**v**

_{f}

**Alfv ´en waves** **Magnetosonic waves**

*•*

Note: **the eigenfunctions are mutually orthogonal:**

### ˆ

**v**

*s*

**⊥ ˆv**

**⊥ ˆv**

*A*

**⊥ ˆv**

**⊥ ˆv**

*f*

*.*

### (43)

*⇒*

Arbitrary velocity field may be decomposed at all times (e.g. at _{t = 0}

) in the three
MHD waves: _{t = 0}

**the initial value problem is a well-posed problem.**

## Dispersion diagrams (schematic)

[ exact diagrams in book: Fig. 5.3, scaling

_{ω ≡ (l/b) ω , ¯k ≡ k l}

_{ω ≡ (l/b) ω , ¯k ≡ k l}

_{¯}

]
b
k_{⊥}

_{ }fast slow ω2 4 5 6 Alfven 7 a k// ω2 fast slow 2 3 Alfven 1 0

*•*

Note: _{ω}

2_{ω}

_{(k}

_{(k}

_{}_{= 0) = 0}

for Alfv ´en and slow waves *⇒*

potential onset of *instability.*

*•*

Asymptotics of _{ω}

2_{ω}

_{(k}

_{(k}

_{⊥}*→∞)*

characterizes *local*behavior of the three waves:

### ⎧

### ⎪

### ⎪

### ⎨

### ⎪

### ⎪

### ⎩

*∂ω/∂k*

*⊥*

*> 0 ,*

*ω*

*2*

_{f}*→ ∞*

for fast waves,
*∂ω/∂k*

*⊥*

*= 0 ,*

*ω*

*2*

_{A}*→ k*

*2*

_{}*b*

2 for Alfv ´en waves,
*∂ω/∂k*

*⊥*

*< 0 ,*

*ω*

*2*

_{s}*→ k*

*2*

_{}*b*

2

*c*

2
*b*

2 *+ c*

2 for slow waves.
## Phase and group velocity

Dispersion equation

_{ω = ω(k) ⇒}

two fundamental concepts:

_{ω = ω(k) ⇒}1. A *single plane wave* propagates in the direction of

**k**

with the **phase velocity**

**v**

ph *≡*

*ω*

*k*

**n**

*,*

**n ≡ k/k = (sin ϑ, 0, cos ϑ) ;**

**n ≡ k/k = (sin ϑ, 0, cos ϑ) ;**

### (45)

*⇒*

MHD waves are non-dispersive (only depend on angle _{ϑ}

, not on _{ϑ}

**|k|**

):
**|k|**

**(v**

ph### )

_{A}**≡ b cos ϑ n ,**

**≡ b cos ϑ n ,**

### (46)

**(v**

ph### )

_{s,f}*≡*

1
2*(b*

2 *+ c*

2### )

*1 ±*

*1 − σ cos*

2**ϑ n ,**

**ϑ n ,**

*σ ≡*

*4b*

2
*c*

2
*(b*

2 *+ c*

2### )

2*.*

### (47)

2. A *wave packet* propagates with the **group velocity**

**v**

gr *≡*

*∂ω*

**∂k**

**∂k**

*≡*

*∂ω*

*∂k*

*x*

**e**

*x*

### +

*∂ω*

*∂k*

*y*

**e**

*y*

### +

*∂ω*

*∂k*

*z*

**e**

*z*

### ;

### (48)

*⇒*

MHD caustics in directions **b**

, and mix of **n**

and **t**

(**⊥ n**

):
**⊥ n**

**(v**

gr### )

_{A}**= b ,**

**= b ,**

### (49)

**(v**

gr### )

_{s,f}*= (v*

ph### )

_{s,f}**n ±**

**n ±**

*σ sin ϑ cos ϑ*

### 2

*√*

*1 − σ cos*

2*ϑ*

*1 ±*

*√*

*1 − σ cos*

2 *ϑ*

** t**

*.*

### (50)

## Wave packet

Wave packet of plane waves satisfying dispersion equation

_{ω = ω(k)}

:

_{ω = ω(k)}### Ψ

*i*

**(r, t) =**

**(r, t) =**

### 1

*(2π)*

*3/2*

_{∞}*−∞*

*A*

*i*

**(k) e**

**(k) e**

**i(k·r−ω(k)t)**

_{d}

3_{d}

_{k .}

_{k .}

_{(51)}

Evolves from initial shape given by *Fourier synthesis,*

### Ψ

*i*

**(r, 0) =**

**(r, 0) =**

### 1

*(2π)*

*3/2*

_{∞}*−∞*

*A*

*i*

**(k) e**

**(k) e**

**ik·r**

_{d}

3_{d}

_{k ,}

_{k ,}

_{(52)}

where amplitudes

_{A}

_{A}

_{i}_{(k)}

are related to initial values _{(k)}

_{Ψ}

_{i}_{(r, 0)}

by

_{(r, 0)}*Fourier analysis,*

*A*

*i*

**(k) =**

### 1

*(2π)*

*3/2*

_{∞}*−∞*

### Ψ

*i*

**(r, 0) e**

**(r, 0) e**

**−ik·r**_{d}

3
_{d}

*r .*

### (53)

MHD:

_{Ψ}

*– perturbations (*

_{i}_{ρ}

_{ρ}

_{˜}

1*,*

) **v**

### ˜

1 (,*˜e*

1, **B**

### ˜

1);*A*

*– Fourier amplitudes (*

_{i}*ρ*

### ˆ

1*,*

) **v**

### ˆ

1 (,*ˆe*

1, **B**

### ˆ

1).*Example: Gaussian wave packet of harmonics centered at some wave vector*

**k**

_{0},

*A*

*i*

**(k) = ˆ**

*A*

*i*

*e*

*−*

1

2* |(k−k*0)·a|2

_{,}

_{,}

_{(54)}

corresponds to initial packet with main harmonic

**k**

_{0}and modulated amplitude centered at

**r = 0**

:
### Ψ

*i*

**(r, 0) = e**

**(r, 0) = e**

*0*

**ik**

**·r***×*

### ˆ

*A*

*i*

*a*

_{x}*a*

_{y}*a*

_{z}*e*

*−*1

## Wave packet (cont’d)

For arbitrary wave packet with localized range of wave vectors, we may expand the dis-persion equation about the central value

**k**

_{0}:

**ω(k) ≈ ω**

0 **ω(k) ≈ ω**

**+ (k − k**

0**+ (k − k**

*) ·*

*∂ω*

**∂k**

**∂k**

**k**0

*,*

*ω*

0 **≡ ω(k**

0**≡ ω(k**

*) .*

### (56)

Inserting this approximation in the expression (51) for the wave packet gives### Ψ

*i*

**(r, t) ≈ e**

**(r, t) ≈ e**

*0*

**i(k***0*

**·r−ω***t)*

*×*

### 1

*(2π)*

*3/2*

_{∞}*−∞*

*A*

*i*

**(k) e**

**(k) e**

* i(k−k*0)·(r−(∂ω/∂k)

_{k0}*t)*

_{d}

3_{d}

_{k ,}

_{k ,}

_{(57)}

representing a carrier wave

_{exp i(k}

_{exp i(k}_{0}

**· r − ω**

**· r − ω**

_{0}

_{t)}

with an amplitude-modulated envelope.
Through constructive interference of the plane waves, _{t)}

*the envelope maintains its shape*during an extended interval of time, whereas

*the surfaces of constant phase of the envelope*

*move with the group velocity,*

**v**

gr ### =

**dr**

**dr**

*dt*

*const. phase*

### =

*∂ω*

**∂k**

**∂k**

**k**0

*,*

### (58)

**
**

## Example: Alfv ´en waves

b
a
b
**v**_{ph}
ϑ
c
b
- b

(a) Phase diagram for Alfv ´en waves is circle

*⇒*

(b) wavefronts pass through points *±b*

## Group diagram: queer behavior

*•*

*Group diagrams with*

**v**

_{gr}

*relative to*

**n**

*for*

*the three MHD waves in the first quadrant.*Group velocities exhibit

*mutually exclusive*

*directions of propagation:*When

**n**

goes
from _{ϑ = 0}

(_{ϑ = 0}

** B**

) to **B**

_{ϑ = π/2}

(_{ϑ = π/2}

**⊥ B**

),
the fast group velocity changes from
par-allel to perpendicular (though it does not
remain parallel to **⊥ B**

**n**

), the Alfv ´en group
ve-locity remains purely parallel, but the slow
*group velocity initially changes clockwise*from parallel to some negative angle and then back again to purely parallel. In the perpendicular direction,

*slow wave*

*pack-ages propagate opposite to direction of*

**n**

*!*- . 5

_{ . 0}

_{ . 5}

_{ 1 . 0}

. 0
1 . 0
1 . 5
v_{z}
⎯
v_{x}
⎯
c/b = 0.8
o
o
o
o
o
onn
o
o
o
o
o
o
o
o
o
o
o
s
o
o
o
o
o o
o
o
o
o
o f
o
o
o
o
o
A

**
**

## Friedrichs diagrams (schematic)

[ exact diagrams in book: Fig. 5.5, parameter_{c/b =}

1_{c/b =}

_{2}

_{γβ , β ≡ 2p/B}

2]
b
_{γβ , β ≡ 2p/B}

**B**Alfvén b c slow fast

**•**

**•**

bc
s A
f
b c
**B**

**n**slow Alfvén fast a 2 2 b + c 2 2 b + c 2 2 b + c

**Phase diagram** **Group diagram**

- 1 . 0 0 . 0 1 . 0
- 2 . 0
- 1 . 0
1 . 0
2 . 0
v_{z}
⎯
v_{x}
⎯
- 1 . 0 0 . 0 1 . 0
- 2 . 0
- 1 . 0
0 . 0
1 . 0
2 . 0
v_{z}
⎯
- 1 . 0 0 . 0 1 . 0
- 2 . 0
- 1 . 0
0 . 0
1 . 0
2 . 0
v_{z}
⎯
( b ) G r o u p d i a g r ams
- 1 . 0 0 . 0 1 . 0
- 2 . 0
- 1 . 0
1 . 0
2 . 0
v_{z}
⎯
v_{x}
⎯
c/b = 0.8
o
o
oo
- 1 . 0 0 . 0 1 . 0
- 2 . 0
- 1 . 0
0 . 0
1 . 0
2 . 0
v_{z}
⎯
c/b = 1.0
o
o
oo
- 1 . 0 0 . 0 1 . 0
- 2 . 0
- 1 . 0
0 . 0
1 . 0
2 . 0
v_{z}
⎯
c/b = 1.2
oo
o
o

**
**

## Summary

*•*

[*Entropy waves:*non-propagating density / entropy perturbations; ]

*•*

*Alfv ´en waves:*incompressible velocity perturbations

*⊥*

plane of **k & B**

, preferably
propagating ** B**

;
**B**

*•*

*Fast magnetoacoustic waves:*compressible velocity perturbations in the plane of

**k & B**

, generalization of sound waves with contributions of the magnetic pressure,
propagating in all directions but fastest **⊥ B**

;
**⊥ B**

*•*

*Slow magnetoacoustic waves:*compressible velocity perturbations in plane of

**k & B**

,
kind of sound waves with impeded propagation **⊥ B**

(orthogonal to fast modes).
**⊥ B**

**
**

## Connection with next subject

Group diagram has a much wider applicability than just wave propagation in infinite ho-mogeneous plasmas: Construction of wave packet involves contributions of large

**k**

(small
wavelengths) so that the **concept of group velocity is essentially a local one. It**re-turns in

*non-linear MHD of inhomogeneous plasmas, where the associated concept of*

**characteristics**describes the propagation of initial data information through the plasma.

## Method

*•*

*Linear advection equation*in one spatial dimension with unknown

_{Ψ(x, t)}

,
_{Ψ(x, t)}

*∂Ψ*

*∂t*

*+ u*

*∂Ψ*

*∂x*

*= 0 ,*

### (59)

and given advection velocity

_{u}

. For _{u}

_{u = const}

, the solution is trivial:
_{u = const}

*Ψ = f(x − ut) ,*

*where f = Ψ*

0 *≡ Ψ(x, t = 0) .*

### (60)

*⇒*

Initial data _{Ψ}

_{0}propagate along

*characteristics:*parallel straight lines

_{dx/dt = u}

.
_{dx/dt = u}

*•*

For _{u}

not constant, characteristics become solutions of the ODEs
_{u}

*dx*

*dt*

*= u(x, t) .*

### (61)

Along these curves, solution

_{Ψ(x, t)}

of (59) is
const:
_{Ψ(x, t)}

### dΨ

*dt*

*≡*

*∂Ψ*

*∂t*

### +

*∂Ψ*

*∂x*

*dx*

*dt*

*= 0 .*

### (62)

*⇒*

For given initial data, the solution can be
determined at any time _{t}

_{t}

_{1}

_{> 0}

by constructing
_{> 0}

*characteristics through suitable set of points.*E.g.,

_{Ψ(x}

_{Ψ(x}

_{i}_{, t}

_{, t}

_{1}

_{) = Ψ}

_{0}

_{(x}

_{(x}

_{i}_{)}

for ‘tent’ function.
i
t
x
t_{1}
x
x
x_{i-1} _{i} _{i+1}
x ...

**
**

## Method (cont’d)

*•*

The method of characteristics generalizes to *nonlinear*partial differential equations: basis of modern developments in

*computational (magneto-)fluid dynamics*[C(M)FD].

*•*

Example: *Quasi-linear advection equation*when

_{u}

is also a function of the unknown _{u}

_{Ψ}

itself. With _{u = Ψ}

, we obtain _{u = Ψ}

*Burgers’ equation:*

*∂Ψ*

*∂t*

### + Ψ

*∂Ψ*

*∂x*

*= ν*

*∂*

2### Ψ

*∂x*

2 *,*

### (63)

where viscous RHS models balance between nonlinear and dissipative processes. At first neglecting this small term, the characteristics are the solutions of the ODE

*dx*

*dt*

*= Ψ(x(t), t) ,*

### (64)

which are just a set of straight lines with slopes determined by the initial data. For large
times, the characteristics will cross, but the build-up of large gradients is counteracted
by smoothing through the dissipative RHS term. This occurs in a very narrow region,
so that a valid solution with a *shock* is obtained in the limit

_{ν → 0}

.
_{ν → 0}

## Classification of PDEs

*•*

Quasi-linear second order PDE in two dimensions:
*A(Φ*

*x*

*, Φ*

*y*

*, x, y) Φ*

*xx*

*+ 2B(...) Φ*

*xy*

*+ C(...) Φ*

*yy*

*= D(...) ,*

### (65)

With

_{Ψ}

_{1}

*≡ Φ*

_{x}_{, Ψ}

_{, Ψ}

_{2}

*≡ Φ*

_{y}*⇒*

equivalent system of first order equations:
*A(Ψ*

1*, Ψ*

2*, x, y) Ψ*

_{1x}*+ B(...) Ψ*

_{1y}*+ B(...) Ψ*

_{2x}*+ C(...) Ψ*

_{2y}*= D(...) ,*

### (66)

### Ψ

*1y*

*− Ψ*

*2x*

*= 0 .*

*Cauchy problem:* find

_{Ψ}

_{1}and

_{Ψ}

_{2}away from boundary

_{C}

when they are given on it.
_{C}

*•*

Replace coordinates _{x, y}

by boundary fitted
coordinates _{x, y}

_{ξ, η}

, where boundary _{ξ, η}

_{C}

is given
by _{C}

_{ξ(x, y) = ξ}

_{ξ(x, y) = ξ}

_{0}. Boundary data become

### Ψ

1*(ξ*

0*, η) = f*

1*(η) ,*

### Ψ

2*(ξ*

0*, η) = f*

2*(η) .*

### (67)

We wish to investigate under which condi-tions

_{Ψ}

_{1}

_{(ξ, η)}

and _{(ξ, η)}

_{Ψ}

_{2}

_{(ξ, η)}

may be obtained
by means of a power series solution about a
particular point _{(ξ, η)}

_{(ξ}

_{(ξ}

_{0}

_{, η}

_{, η}

_{0}

_{)}

on the boundary.
y
x
C
η_{1}
η_{2}
η_{-1}
ξ_{2}
ξ_{1}
ξ_{0}
η_{0}
o_{o}o**ooo**

**
**

## Classification (cont’d)

*•*

Power series:
### Ψ

1*(ξ, η) =*

### Ψ

1*(ξ*

0*, η*

0### )

*+ (ξ − ξ*

0### )

*∂Ψ*

1
*∂ξ*

0 *+ (η − η*

0### )

*∂Ψ*

1
*∂η*

0 *+ · · · ,*

### (68)

### Ψ

2*(ξ, η) =*

### Ψ

2*(ξ*

0*, η*

0### )

*+ (ξ − ξ*

0### )

*∂Ψ*

2
*∂ξ*

0 *+ (η − η*

0### )

*∂Ψ*

2
*∂η*

0 *+ · · · .*

Green expressions known from boundary conditions (67)

*⇒*

we need to investigate
under which circumstances remaining expressions _{(∂Ψ}

_{(∂Ψ}

_{i}_{/∂ξ)}

_{/∂ξ)}

_{0}can be calculated.

*•*

Transform PDEs (66) to _{ξ}

-_{ξ}

_{η}

coordinates:
_{η}

*(Aξ*

*x*

*+ Bξ*

*y*

### )

*∂Ψ*

1
*∂ξ*

*+ (Bξ*

*x*

*+ Cξ*

*y*

### )

*∂Ψ*

2
*∂ξ*

*= D − (Aη*

*x*

*+ Bη*

*y*

### )

*∂Ψ*

1
*∂η*

*− (Bη*

*x*

*+ Cη*

*y*

### )

*∂Ψ*

2
*∂η*

*, (69)*

*ξ*

*y*

*∂Ψ*

1
*∂ξ*

*− ξ*

*x*

*∂Ψ*

2
*∂ξ*

*= −η*

*y*

*∂Ψ*

1
*∂η*

*+ η*

*x*

*∂Ψ*

2
*∂η*

*.*

The unknown derivatives

_{∂Ψ}

_{∂Ψ}

_{1}

_{/∂ξ}

and _{/∂ξ}

_{∂Ψ}

_{∂Ψ}

_{2}

_{/∂ξ}

may be determined from Eqs. (69)
_{/∂ξ}

**if the determinant of the coefficients on the left hand side does not vanish.**

## Classification (cont’d)

*•*

Vice versa, condition that the determinant vanishes,
*Aξ*

*x*

*+ Bξ*

*y*

*Bξ*

*x*

*+ Cξ*

*y*

*ξ*

*y*

*−ξ*

*x*

* = −Aξ*

2
*x*

*− 2Bξ*

*x*

*ξ*

*y*

*− Cξ*

2
*y*

*= 0 ,*

### (70)

defines two directions in every point of the plane, *the characteristic directions,* **along**
**which posing Cauchy boundary conditions does not determine the solution:**

*dy*

*dx*

char *= −*

*ξ*

*x*

*ξ*

*y*

### =

*B ±*

*√*

*B*

2 *− AC*

*A*

*.*

### (71)

*•*

Three cases:
(a)

_{B}

2 _{B}

_{> AC ⇒}

characteristics are real: _{> AC ⇒}

*hyperbolic*equation

(example: wave equation

### Φ

_{xx}*− (1/c*

2### )Φ

_{tt}### = 0

);(b)

_{B}

2 _{B}

_{= AC ⇒}

characteristics are real but coincide: _{= AC ⇒}

*parabolic*equation

(example: heat equation

### Φ

_{xx}*− (1/λ)Φ*

_{t}### = 0

);(c)

_{B}

2 _{B}

_{< AC ⇒}

characteristics are complex: _{< AC ⇒}

*elliptic*equation

**
**

## Apply to MHD equations

*•*

Instead of 2-vector _{(Ψ}

_{1}

_{, Ψ}

_{, Ψ}

_{2}

_{)}

: 8-vector _{Ψ}

_{i}_{(i = 1, · · · 8)}

for variables _{(i = 1, · · · 8)}

_{ρ}

, _{ρ}

**v**

, _{e}

, _{e}

**B (r, t)**

.
**B (r, t)**

*•*

We will prove: MHD equations are *symmetric hyperbolic*PDEs; they posses complete set of

*real characteristics related to the eigenvalues of the linearized system.*

*•*

Apply same method as before: Assume boundary data for _{ρ}

, _{ρ}

**v**

, _{e}

, _{e}

**B**

to be given on a
3-dimensional manifold in 4-dimensional space-time **r, t**

:
**r, t**

**ξ(r, t) = ξ**

0**ξ(r, t) = ξ**

*.*

### (72)

(Visualize as being swept out by motion of 2-D surfaces in ordinary 3-D space

**(r)**

when
time

_{t}

progresses.)
_{t}

*•*

**Duality: – If this manifold is characteristic**

*⇒*

Cauchy problem ill-posed on it;
– If this manifold is not characteristic *⇒*

Cauchy problem well-posed on it.
*•*

Hence, for IVP in MHD (where **ρ(r, 0)**

, **ρ(r, 0)**

**v(r, 0)**

, **v(r, 0)**

**e(r, 0)**

, **e(r, 0)**

**B(r, 0)**

are given on domain in
**B(r, 0)**

ordinary 3-space) to be well-posed, ordinary 3-space should not be a characteristic.

*•*

We will prove that the characteristics in MHD are real 3-dimensional manifolds involving
time, so that *the IVP in MHD is well-posed.*

## Application (cont’d)

*•*

Cover 4-space _{(r, t)}

by boundary-fitted coordinates

_{(r, t)}_{ξ}

, _{ξ}

_{η}

, _{η}

_{ζ}

, _{ζ}

_{τ}

, and try power series:
_{τ}

*ρ(ξ, η, ζ, τ ) =*

*ρ*

0*(η*

0*, ζ*

0*, τ*

0### )

*+ (ξ − ξ*

0### )

*∂ρ*

*∂ξ*

0 *+ (η − η*

0### )

*∂ρ*

*∂η*

0
*+ (ζ − ζ*

0### )

*∂ρ*

*∂ζ*

0 *+ (τ − τ*

0### )

*∂ρ*

*∂τ*

0 *+ · · ·*

etc. for **v**

, _{e}

, _{e}

**B**

_{. (73)}

_{. (73)}

*•*

Problem solvable if unknowns _{(∂ρ/∂ξ)}

_{(∂ρ/∂ξ)}

_{0},

_{(∂v/∂ξ)}

_{(∂v/∂ξ)}_{0},

_{(∂e/∂ξ)}

_{(∂e/∂ξ)}

_{0},

_{(∂B/∂ξ)}

_{(∂B/∂ξ)}_{0}can be constructed from MHD equations. Indicate those by a prime:

*∇f = ∇ξ*

*f*

*+ ∇η*

*∂f*

*∂η*

*+ ∇ζ*

*∂f*

*∂ζ*

*+ ∇τ*

*∂f*

*∂τ*

*,*

_{(74)}

*Df*

*Dt*

*= (ξ*

*t*

**+ v · ∇ξ)**

**+ v · ∇ξ)**

*f*

_{+ (η}

_{+ (η}

*t*

**+ v · ∇η)**

**+ v · ∇η)**

*∂f*

*∂η*

*+ (ζ*

*t*

**+ v · ∇ζ)**

**+ v · ∇ζ)**

*∂f*

*∂ζ*

*+ (τ*

*t*

**+ v · ∇τ )**

**+ v · ∇τ )**

*∂f*

*∂τ*

*.*

*•*

Translation recipe (similar to shock recipe of Sec.4.5):
**∇f → n**

**∇f → n**

*f*

### +

*· · ·*

*,*

**n ≡ ∇ξ :**

**n ≡ ∇ξ :**

*normal to the characteristic*

_{,}

_{,}

### (75)

*Df*

*Dt*

*→ −u*

*f*

*
*

_{+}

_{· · ·}

_{· · ·}

_{, −u ≡ ξ}

_{, −u ≡ ξ}

**
**

## Application (cont’d)

*•*

This gives:
*− u*

*ρ*

**+ ρ n ·**

**+ ρ n ·**

**v**

### =

*· · ·*

*,*

*− ρuv*

**+ (γ − 1) n (e**

**+ (γ − 1) n (e**

*ρ*

*+ ρ*

*e*

**) + (n B · −n · B)**

**) + (n B · −n · B)**

**B**

### =

*· · ·*

*,*

### (76)

*− u*

*e*

**+ (γ − 1)e n ·**

**+ (γ − 1)e n ·**

**v**

### =

*· · ·*

*,*

*− uB*

**+ (B n · −n · B)**

**+ (B n · −n · B)**

**v**

### =

*· · ·*

*,*

**n ·**

**n ·**

**B**

### =

*· · ·*

*.*

*LHS analogous to EVP (30) for linear MHD waves,* where

**k → n**

and **k → n**

_{ω → u}

!
_{ω → u}

*•*

**Duality:**– Values

_{ρ}

_{ρ}

_{, v}

_{, v}

_{, e}

_{, e}

_{, B}

_{, B}*may not be found if*

*Δ ≡ u(u*

2 *− b*

2

_{n}### )

*u*

4 *− (b*

2 *+ c*

2*)u*

2 *+ b*

2

_{n}*c*

2 *= 0 ⇒*

*ξ*

0 characteristic ### ;

### (77)

– Values

_{ρ}

_{ρ}

_{, v}

_{, v}

_{, e}

_{, e}

_{, B}

_{, B}*may be found if*

*Δ = 0 ⇒*

*ξ*

0 not characteristic (solutions may be propagated away from it) *.*

*⇒*

*7 real characteristics*, corresponding to 7 linear waves (entropy, Alfv ´en, slow, fast).

**The equations of ideal MHD are symmetric hyperbolic equations, and**

**the initial**

**value problem is well-posed**(Friedrichs).

## Application (cont’d)

**B**bc 2 2 b +c Alfvén b c slow fast • • • • 2 2 b +c f -x t A-

_{s}-

_{E}

_{s}+

_{A}+

_{f}+

Group diagram is the *ray surface,*
i.e. the spatial part of characteristic
manifold at certain time

_{t}

_{t}

_{0}.

*x*

*-*

_{t}

_{t}

*cross-sections of 7 characteristics*(

_{x}

-axis oblique with respect to _{x}

**B**

;
inclination of entropy mode E indicates plasma background flow).

*•*

*Locality*of group diagrams and characteristics neglects

*global plasma inhomogeneity.*