Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 57 / 3

Use $D={\partial }_t+w{\partial }_z$, $w=v_z$, and a prime for ${\partial }_z$. For the plane-parallel magnetohydrodynamic flow with imposed shear, $\nabla \cdot \mathbf{u}=w^{\prime }$, whileThe solenoidal magnetic field constraint gives $B_z^{\prime }=0$. The $z$ component of the MHD induction equation then gives ${\partial }_t{B}_z=0$. Thus $B_z$ is constant in both space and time.

Use the magnetic tension and magnetic pressure decompositionThe continuity equation and momentum equation now reduce toThe transverse components of the MHD induction equation areThe terms involving $a$ respectively accelerate the flow across the imposed shear flow and wind the transverse magnetic field. The closure is the isothermal equation of state $p=c_s^2\rho$, as printed in the PDF.

The logarithmic density term is the barotropic fluid energy for $p=c_s^2\rho$, rather than the thermodynamic heat content of an isolated gas. DefineFor any scalar $q$, the continuity equation implies ${\partial }_t(\rho q)+{\partial }_z(\rho wq)=\rho Dq$. Therefore $De=-c_s^2{w}^{\prime }$, $D\Phi =wg$, and dotting the momentum equations with $\rho \mathbf{v}$ givesThe MHD induction equation gives the complementary magnetic energy balanceAdding these two identities, and collecting the product derivatives, yields the barotropic magnetic energy equationHere $E=E_h+E_B$. The two stress terms in $S$ describe magnetohydrodynamic shear work: the maintained background shear flow can supply energy to, or remove energy from, the perturbation flow and magnetic field. The sign depends on the off-diagonal total stress; this is why the energy excluding the background shear flow is not generally conserved. Changing ${\rho }_0$ merely adds a constant multiple of the conserved density to $E$ and the corresponding mass flux to $F$, leaving $S$ unchanged.

For a steady flow, mass conservation gives $\rho w=\text{constant}$ and ${\rho }^{\prime }/\rho =-w^{\prime }/w$. Define the Alfvén velocity ${\mathbf{v}}_a=\mathbf{B}/\sqrt{{\mu }_0\rho }$ and $C=w^2-v_{az}^2$. Combining the horizontal momentum and MHD induction equations without dividing by $C$ givesDot these identities with $B_x/({\mu }_0\rho )$ and $B_y/({\mu }_0\rho )$, respectively:The vertical momentum equation and the isothermal equation of state giveMultiply by $C$ and eliminate the magnetic derivative to obtainNo division by $w^2-v_{az}^2$ was needed in deriving this necessary relation.

The magnetosonic critical speeds in the $z$ direction areThe plus sign gives the fast magnetosonic wave speed and the minus sign the slow magnetosonic wave speed. Thus the differential coefficient is $(w^2-c_{\mathrm{f}}^2)(w^2-c_{\mathrm{s}}^2)$. A smooth outflow proceeding from below both speeds to above both must normally pass through both magnetosonic critical speeds. At each crossing, the right-hand side must also vanish; this is the regularity at a magnetosonic point condition. The derivative coefficient changes sign at each nondegenerate crossing. For $w>0$ and $w^{\prime }>0$, the driving term must be positive below the slow point, negative between the points and positive above the fast point.

The Alfvén speed component satisfies $c_{\mathrm{s}}^2\le v_{az}^2\le c_{\mathrm{f}}^2$. With downward gravity $g>0$, the gravitational term is consequently nonnegative at the slow point and nonpositive at the fast point. The magnetohydrodynamic shear work contribution must balance it at each point, and can provide the upward driving needed to pass the fast point. For $a=0$ and strictly positive $g$, generic separated slow and fast points cannot satisfy the required zero numerator; special vanishing-gravity or coincident-speed cases need separate treatment.

There is also Alfvén-point compatibility in a plane-parallel sheared flow. At $w^2=v_{az}^2$, the original transverse equations requireThese restrictions are not generally visible as a zero of the scalar differential coefficient, which there equals $-v_{az}^2(v_{ax}^2+v_{ay}^2)$. In particular, a strictly accelerating regular solution must have $B_x=0$ at that point. The scalar relation is therefore a necessary wind equation, not a substitute for regularity of all the original ideal magnetohydrodynamic equations. Degenerate cases such as a purely longitudinal magnetic field can merge characteristic speeds and reduce the number of distinct critical conditions.