Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 51 / 2

Use the Galerkin method, rather than pointwise equality: products generate harmonics outside the retained five-dimensional space. Put $X=\alpha x$, $Z=\pi z$, and denote the five prefactors byThe retained streamfunction is $P\sin X\sin Z$, the retained temperature perturbation is $T\cos X\sin Z+C\sin 2Z$, and the retained magnetic-flux perturbation is $D\sin X\cos Z+E\sin 2X$. These satisfy all the stated boundary conditions. Let $J(f,g)=f_x{g}_z-f_z{g}_x$, the streamfunction advection bracket. Orthogonality of the sine and cosine modes gives the following coefficients of the retained harmonics:For example the $C$ contribution to the temperature bracket is $\alpha \pi PC[\cos X\sin 3Z-\cos X\sin Z]$. The magnetic bracket follows by writing $A=A_D+A_E$: $J(A,\Delta A)=({\beta }^2-4{\alpha }^2)J(A_D,A_E)$ and $J(A_D,A_E)=\alpha \pi DE(\sin 3X-\sin X)\sin Z$. These identities exhibit both the retained terms and discarded harmonics.

Before rescaling time, the projected coefficient equations areSubstitute the prefactors and divide time derivatives by ${\beta }^2$. With $\tau ={\beta }^{2}t$, $\varpi =4{\pi }^2/{\beta }^2$ and $q={\pi }^{2}Q/{\beta }^4$, the resulting five-mode vertical-field magnetoconvection equations areHere primes denote differentiation in $\tau$. The factor $3-\varpi$ is $(3{\alpha }^2-{\pi }^2)/{\beta }^2$, which checks the magnetic nonlinear sign. There is a terminology qualification in the source: on this thermal diffusion time scale, its $\zeta$ is the magnetic-to-thermal diffusivity ratio $\eta /\kappa$. The conventional magnetic Prandtl number is $\nu /\eta =\sigma /\zeta$, where $\sigma$ is the Prandtl number. The displayed equations and subsequent calculation use the source's $\zeta$ unchanged.

At conduction, $c$ and $e$ have stable eigenvalues $-\varpi$ and $-(4-\varpi )\zeta$. The remaining linearization on $(a,b,d)$ isIts characteristic polynomial is $m^3+A{m}^2+Bm+C$, whereConsequently the steady threshold is $\boxed{r_s=1+q}$. The symmetry $(a,b,d)\mapsto (-a,-b,-d)$, with $(c,e)$ unchanged, gives a generic pitchfork bifurcation, whose forward or backward direction is determined below. At threshold the other two critical-block eigenvalues satisfy $m^2+Am+B_s=0$, where $B_s=\zeta (\sigma +1)-\sigma (1-\zeta )q$. They are stable if $B_s>0$; if $B_s<0$ there is already one unstable transverse direction. Degenerate zero cubic slope requires higher-order analysis and is not a generic pitchfork.

At a Hopf bifurcation, $AB=C$, $B={\omega }^2>0$. Solving the coefficient equation givesFor physical $q\ge 0$ the oscillatory threshold therefore exists exactly when $0<\zeta <1$ and $q>q_{\ast }$. The conduction double-zero criterion for five-mode magnetoconvection givesAt this Bogdanov–Takens bifurcation, the polynomial is $m^2(m+A)$ and the zero eigenspace is one-dimensional. For $q<q_{\ast }$ conduction first loses stability at the steady threshold; for $q>q_{\ast }$ it first loses stability at $r_H<r_s$. For $\zeta \ge 1$ no positive-$q$ double-zero or oscillatory threshold exists. These conclusions also follow from the Routh-Hurwitz stability criterion, $B,C>0$ and $AB>C$. The Hopf bifurcation is transversal since ${\partial }_r(AB-C)=-\sigma (\sigma +1)\ne 0$; its nonlinear criticality is not required by this linear onset calculation.

On the nonzero stationary branch of five-mode magnetoconvection, put $s=a^2$ and $m=(4-\varpi ){\zeta }^2/\varpi >0$. The temperature equations first give $b=a/(1+s)$ and $c=s/(1+s)$. The magnetic equations then giveDivide the steady first equation by $\sigma a$ and substitute these expressions. The exact branch isExpanding at $s=0$ givesNear $\varpi =2$ the required amplitude restriction $a^2\ll {\zeta }^2$ ensures $s\ll m$. For $\varpi \le 2$, $r_2>0$, so the paired equilibrium points emerge in the forward direction $r>1+q$. For fixed $q>0$ and small $\zeta$, the pitchfork reversal near geometric ratio two in magnetoconvection occurs atThis follows by expanding the exact zero-slope equation $q\varpi (2-\varpi )+(4-\varpi ){\zeta }^2(1+q)=0$. Above this value the branch initially runs backwards into $r<1+q$; at $q=0$ there is no reversal. This is a degenerate pitchfork bifurcation at $r_2=0$, not a conclusion that every emerging state is stable. If $B_s>0$, the center-direction cubic term has the opposite sign to $r_2$: the forward branch is locally attracting and the backward branch repelling. If $B_s<0$, an unstable transverse mode remains.

For the nearby geometry, the next coefficient is $r_4=q[(2\varpi -5)/m^2+(2-\varpi )/m]$, which is negative near the reversal. Thus a small positive $r_2$ produces a nearby maximum of $r$ at $s\simeq -r_2/(2r_4)>0$; this fold approaches the pitchfork as $r_2\downarrow 0$. Once $r_2<0$ the small-amplitude branch is backward. The exact formula eventually grows like $r\sim s$, so a backward branch must turn at a finite amplitude, potentially outside the small-amplitude expansion.