Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 69 / 1

For an axisymmetric magnetic field, the solenoidal condition becomes ${\partial }_R(R{B}_R)+{\partial }_z(R{B}_z)=0$. Locally introduce a poloidal magnetic flux function $\Psi$ byThe divergence condition is exactly the integrability condition for these two derivatives. The additive constant in $\Psi$ fixes the reference flux. For a field regular on the axis choose $\Psi (0,z,t)=0$. Then the magnetic flux through a horizontal disk of radius $R$ is $2\pi \Psi (R,z,t)$, and the flux between two such surfaces is $2\pi \Delta \Psi$.

Since ${\mathbf{B}}_p\cdot \nabla \Psi =0$ and ${\partial }_{\varphi }\Psi =0$, the level surfaces of $\Psi$ contain the complete magnetic field, including its toroidal magnetic field component. Thus regular levels are poloidal flux surfaces. Their meridional contours trace the projected field lines. The representation is local on regular regions; disconnected flux surfaces with the same label need not share additional physical data.

The azimuthal component of $\mathbf{u}\times \mathbf{B}$ is $u_z{B}_R-u_R{B}_z=-{\mathbf{u}}_p\cdot \nabla \Psi /R$. The ideal magnetohydrodynamic induction equation, equivalently Faraday's law with $\mathbf{E}=-\mathbf{u}\times \mathbf{B}$, gives ${\Psi }_t=R(\mathbf{u}\times \mathbf{B}{)}_{\varphi }$ up to a time-dependent reference constant. Fixing that reference gives the material advection of an axisymmetric magnetic flux function:Axisymmetry removes the azimuthal contribution to the material derivative. This is magnetic flux freezing expressed using one scalar flux label.

The azimuthal component itself is generally not an advected scalar. Including the cylindrical basis-vector derivatives in the induction equation givesUsing mass conservation, $D\rho /Dt=-\rho \nabla \cdot \mathbf{u}$, and $DR/Dt=u_R$, this becomes the toroidal magnetic-field winding equation:Thus $B_{\varphi }/(\rho R)$ is conserved following the flow if the angular velocity is constant along the poloidal magnetic field, including the case of no rotation. Otherwise differential rotation winds poloidal field into toroidal field.

There is also an explicit flux-conservation form,Integrating this over a meridional region converts its change of toroidal flux to boundary terms. Hence such a flux is conserved when the indicated boundary transport vanishes. More generally magnetic flux freezing conserves flux through every material surface, but a meridional coordinate surface need not remain material when the fluid rotates. Toroidal flux conservation therefore does not imply that $B_{\varphi }$ itself is materially constant.

Use the standard uniform-density incompressible flow version of ideal magnetohydrodynamics, in which $\nabla \cdot \mathbf{u}=0$ and pressure enforces that constraint. The finite-$\gamma$ compressible adiabatic pressure equation is replaced by this pressure constraint. The consequences of retaining that cover-sheet equation literally as an additional condition are given below.

Write $F=f(R,z+v(R)t)$, $G=g(R,z-v(R)t)$, $U=F+G$ and $b=F-G$, where $b$ is the toroidal Alfvén velocity. The azimuthal velocity is $U$, while the Alfvén velocity vector is $b{\mathbf{e}}_{\varphi }+v{\mathbf{e}}_z$. Both solenoidal constraints hold automatically. Since there is no meridional flow, the azimuthal magnetohydrodynamic momentum equation and ideal magnetohydrodynamic induction equation reduce toDifferentiating the traveling arguments verifies both equations for arbitrary smooth $f$, $g$ and $v$. In particular, no condition $v^{\prime }=0$ is needed.

The remaining force balance is essential. Define the total pressure per unit mass, including the gravitational potential, byThe radial acceleration is $-U^2/R$, and magnetic curvature tension contributes $-b^2/R$. The radial and axial momentum equations are consequentlyThey are compatible if and only ifThis is the pressure compatibility of nonlinear torsional Alfvén profiles. It is also sufficient locally: when $FG$ is independent of $z$, chooseA stationary axisymmetric $\Phi$ satisfying the Poisson equation can be absorbed in this pressure. Gravity need not vanish. If the axis is included, smoothness requires $U,b=O(R)$ and a regular axial field; $f,g=O(R)$ and smooth even $v$ give the usual regular extension. The pressure constant may be chosen to make $p$ positive on a bounded region where the other terms are bounded.

The compatible profiles have the following interpretation. At a cylinder with $v\ne 0$, the variables $a=z+vt$ and $c=z-vt$ are independent as $z,t$ vary. Compatibility reads $f_a(R,a)g(R,c)+f(R,a)g_c(R,c)=0$. If both profiles are nonzero, division gives $f_a/f=-g_c/g=\kappa (R)$, independent of both traveling arguments. Thereforelocally on such a region. Alternatively, one profile vanishes identically and the other is unrestricted. Choosing a point where one profile is nonzero shows that a nontrivial opposite profile obeys the same linear ordinary differential equation everywhere on a connected axial interval, so isolated zeros do not create additional overlapping branches.

If $v>0$, a pure $f$ profile travels towards negative $z$, and a pure $g$ profile towards positive $z$, each at the local Alfvén speed $v(R)$. Their velocity and toroidal-field perturbations have equal magnitudes, with respectively the same or opposite signs. They are finite-amplitude torsional Alfvén waves. For a single profile, centrifugal force and magnetic curvature tension cancel, leaving spatially constant $\Pi$; gas pressure compensates the varying magnetic pressure. The waveform is transported without axial steepening. Different cylinder speeds shear its phase radially, magnetohydrodynamic phase mixing.

At linear order both wave families may be superposed, since their interaction product is second order in amplitude. Two arbitrary opposing profiles cannot simply be superposed at finite amplitude: their product would generate an axial variation in the radial force imbalance, incompatible with purely azimuthal velocity. The exponential pair is a formal exception, but is unbounded in an axial direction when $\kappa \ne 0$. If both profiles must be bounded on the whole axial line, $\kappa =0$, so $f=A(R)$ and $g=C(R)$ describe a steady rotating, twisted-field equilibrium. Their pressure gradient balances the remaining centrifugal and magnetic curvature forces. At $v=0$ there is no propagation; the condition is just $f(R,z)g(R,z)=h(R)$, giving additional steady profiles. These cases exhaust the local possibilities under the stated ansatz and domain qualifications.

There is a closure distinction in the wording. Incompressible flow is the formal infinite-sound-speed limit, not the simultaneous imposition of finite-$\gamma$ adiabatic compression and an independently fixed density; see Ogilvie's discussion of simplified fluid models, section 2.8. If the cover-sheet pressure equation is nevertheless imposed in addition, then axisymmetry and purely azimuthal motion give $p_t=0$. In addition to $(FG{)}_z=0$, the necessary and sufficient local pressure-gradient conditions, with stationary $\Phi$, areThey follow by requiring the gradients $p_z/\rho =-{\Phi }_z-\frac{1}{2}(b^2{)}_z$ and $p_R/\rho =4FG/R-{\Phi }_R-\frac{1}{2}(b^2+v^2{)}_R$ to be time independent; the remaining spatially uniform pressure variation is removed by $C(t)$.

For a single traveling profile with $v\ne 0$, these extra conditions require its square to be affine in its traveling argument, $f^2=a(R)(z+vt)+b_0(R)$ or $g^2=a(R)(z-vt)+b_0(R)$, with $v(R)a(R)$ constant across a connected radial region. Smooth bounded profiles on the entire axial line then have $a=0$ and are steady. For two nonzero exponential profiles the extra conditions force $\kappa =0$. Thus arbitrary finite-amplitude traveling pulses are valid for the intended incompressible pressure model, but not generally for the overconstrained finite-$\gamma$ closure. The force compatibility derived above makes the distinction explicit.