The divergence-free poloidal field can be represented by the poloidal magnetic flux function

B_{p} = \nabla ψ \times \nabla ϕ = - \frac{1}{r} e_{ϕ} \times \nabla ψ .

(1)

In a steady axisymmetric ideal magnetohydrodynamics flow, the azimuthal component of

\nabla \times (u \times B) = 0

makes

u_{p}

parallel to

B_{p}

. Write

ρ u_{p} = k B_{p} .

(2)

Mass conservation and

\nabla \cdot B = 0

then imply

B \cdot \nabla k = 0

, so the magnetohydrodynamic mass loading

k = k (ψ)

is constant along each magnetic line.

The poloidal part of

u \times B

u \times B = \frac{1}{r} (u_{ϕ} - \frac{k B _{ϕ}}{ρ}) \nabla ψ .

(3)

Its curl vanishes only if its coefficient is a flux function, giving the field-line angular velocity

\frac{u _{ϕ}}{r} - \frac{k B _{ϕ}}{r ρ} = ω (ψ) .

(4)

The azimuthal component of momentum conservation is a divergence of matter and magnetic angular-momentum flux. Dividing its field-line constant by the mass loading yields the magnetohydrodynamic angular-momentum invariant

r u_{ϕ} - \frac{r B _{ϕ}}{μ _{0} k} = ℓ (ψ) .

(5)

The conservative total-energy equation similarly gives the magnetohydrodynamic Bernoulli invariant

\frac{1}{2} ∣ u ∣^{2} + Φ + h - \frac{r ω B _{ϕ}}{μ _{0} k} = ϵ (ψ) .

(6)

Finally, the entropy advection equation and

u_{p} ∥ B_{p}

imply

s = s (ψ)

. Thus

k, ω, ℓ, ϵ

, and

s

are constant along each magnetic field line.

b

 0  0

Solution

 0  0

Introduce the squared poloidal Alfvén number

M_{A}^{2} = \frac{μ _{0} ρ u _{p}^{2}}{B _{p}^{2}} = \frac{μ _{0} k ^{2}}{ρ} .

(1)

Solving the two linear azimuthal invariants gives

u_{ϕ} = \frac{M _{A}^{2} ℓ / r - r ω}{M _{A}^{2} - 1},

(2)

B_{ϕ} = \frac{μ _{0} k ( ℓ / r - r ω )}{M _{A}^{2} - 1} .

(3)

At an Alfvén surface,

M_{A}^{2} = 1

. Smooth passage through the apparent singularity requires both numerators to vanish at the same cylindrical radius

r_{A}

, giving the Alfvén-surface regularity condition for an axisymmetric wind

ℓ = ω r_{A}^{2} .

(4)

The finite limiting values of

u_{ϕ}

and

B_{ϕ}

then follow by l'Hopital's rule from the local variation of

M_{A}

and

r

along the field line; the algebraic invariants alone fix their combination rather than each value separately at the critical point.

c

 0  0

Solution

 0  0

Along the open line,

B_{p} \sim F / r^{2}

and the mass-loading relation with

u_{p} \to u_{\infty}

gives

ρ \sim \frac{k F}{u _{\infty} r ^{2}}, M_{A}^{2} \sim \frac{μ _{0} k u _{\infty}}{F} r^{2} ⟶ \infty.

(1)

The solutions of part b consequently have

u_{ϕ} \sim \frac{ℓ}{r}, B_{ϕ} \sim - \frac{ω F}{u _{\infty} r} .

(2)

The azimuthal Alfvén speed therefore approaches the nonzero constant

∣ v_{A ϕ} ∣ = \frac{∣ B _{ϕ} ∣}{μ _{0} ρ} ⟶ ∣ ω ∣ \frac{F}{μ _{0} k u _{\infty}} .

(3)

For an unconfined outflow it is natural to take

Φ \to 0

and

h \to 0

at infinity; also

u_{ϕ} \to 0

. The magnetic term in the magnetohydrodynamic Bernoulli invariant tends to

- \frac{r ω B _{ϕ}}{μ _{0} k} ⟶ \frac{ω ^{2} F}{μ _{0} k u _{\infty}} .

(4)

Hence the asymptotic energy of a radial magnetohydrodynamic wind is

ϵ = \frac{1}{2} u_{\infty}^{2} + \frac{ω ^{2} F}{μ _{0} k u _{\infty}} .

(5)

 Ancestors (9)

 View article source

 Discussion (0)

New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 314 / 4

 Discussion (0)  Subscribe (1)

 Discussion (0)