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are treated as constants, the upstream shear stress phase lead promotes growth, while the inclined-bed sediment threshold reduces it and the finite saturation length delays sediment transport. There is an unstable band precisely when $b > 0$ : $0 < k < b / (a L_{sat})$ . Within this band

c > 0

, so the growing bedforms migrate downstream. The fastest normal mode satisfies

(k_{m a x} L_{sat})^{3} + 3 k_{m a x} L_{sat} = 2 b / a .

(4)

This last maximization assumes constant

A, B

; if the hydrodynamic coefficients vary with wavenumber, their derivatives also enter the selection condition. When

b \leq 0

and

a > 0

, all nonzero normal modes decay.

Just above the sediment entrainment threshold,

b ≃ τ_{th, 0} (B - 1/ μ_{s}) .

(5)

Thus the common regime

B < 1/ μ_{s}

is stable near onset: the slope penalty overwhelms the hydrodynamic phase lead. This conclusion is conditional on that inequality, not universal. For constant

B > 0

, instability begins only when

τ_{0} > τ_{th, 0} / (μ_{s} B)

as well as

τ_{0} > τ_{th, 0}

. The transport sensitivity

C = χ γ Δ^{γ - 1}

vanishes at onset for

γ > 1

, is finite for

γ = 1

, and is singular for

0 < γ < 1

. Exactly at threshold, the positive-part transport law needs separate treatment; the above linearization with perturbations small compared with

Δ

is unavailable.

Far above the sediment entrainment threshold,

b ≃ τ_{0} B

. If

A, B > 0

, the unstable cutoff tends to

B / (A L_{sat})

, and the dominant wavelength is set by the saturation length. Both growth rate and migration velocity scale with

C τ_{0} \sim χ γ τ_{0}^{γ}

, apart from their length factors. Their numerical values and any detailed dependence on wavenumber still require a flow closure.

For example, prescribing uniform shear stress independently of the bed gives

T = 0

and

σ = - \frac{C τ _{th, 0}}{μ _{s}} \frac{k ^{2}}{1 + ( k L _{sat} ) ^{2}} < 0.

(6)

This equally admissible closure illustrates why instability cannot be asserted from the printed sediment equations alone.

Figure 1.
Conditional bedform growth and migration
.

The plots use constant

a > 0

and three illustrative ratios

b / a

. They show how the sign of the shear phase lead minus the slope correction determines stability; they are not numerical predictions for unspecified flow conditions.

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Take

k > 0

; reflection in

x

supplies the result for negative wavenumber. Since each region has uniform base velocity, the perturbation is an incompressible, irrotational flow with a velocity potential satisfying the Laplace equation. Its decaying outer forms are

ϕ_{+} = A_{+} e^{- k (y - h)} e^{ik x + σ t}, ϕ_{-} = A_{-} e^{k (y + h)} e^{ik x + σ t} .

(1)

For a varicose mode of a planar jet, reflection symmetry makes the inner potential proportional to

cosh (k y)

; for a sinuous mode of a planar jet, it is proportional to

sinh (k y)

At the upper interface, the linearized kinematic boundary condition is

σ η = \partial_{y} ϕ_{+} (h), (σ + ik U) η = \partial_{y} ϕ_{in} (h) .

(2)

The first equation gives

ϕ_{+} (h) = - σ η / k

. The second gives

ϕ_{in} (h) = \frac{σ + ik U}{k} η {coth (kh), tanh (kh), varicose, sinuous .

(3)

The linearized Unsteady Bernoulli equation gives

p^{'} = - ρ (\partial_{t} + U \partial_{x}) ϕ

. Applying pressure continuity therefore produces the equal-density top-hat planar-jet dispersion relation

σ^{2} + F (σ + ik U)^{2} = 0, F = {coth (kh), tanh (kh), varicose, sinuous .

(4)

Solving the quadratic equation gives

σ = \frac{k U}{1 + F} (\pm F - i F) .

(5)

Equivalently,

σ_{var} σ_{sin} = \pm \frac{k U}{2} 1 - e^{- 4 kh} - \frac{ik U}{2} (1 + e^{- 2 kh}), = \pm \frac{k U}{2} 1 - e^{- 4 kh} - \frac{ik U}{2} (1 - e^{- 2 kh}) .

(6)

The plus sign is the unstable Kelvin-Helmholtz instability branch. In particular, the two modes have exactly the same growth rate,

σ_{R} (k) = \frac{k U}{2} 1 - e^{- 4 kh} .

(7)

Its limiting forms are

σ_{R} \sim U h k^{3/2} (kh \to 0), σ_{R} \sim \frac{U k}{2} (kh \to \infty) .

(8)

Thus neither type grows faster in this equal-density idealization, although their phase speeds differ. The unbounded large-

k

growth is the familiar short-wave pathology of an inviscid vortex sheet.

Figure 1.
Growth rate of varicose and sinuous planar-jet disturbances
. The two modes have the same dimensionless growth curve, with the long-wave and short-wave asymptotes shown.

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For a discrete Fourier mode

u_{m}^{n} = r^{n} e^{im θ}

, put

q = 4 μ sin^{2} \frac{θ}{2}, 0 \leq q \leq 4 μ .

(1)

The recurrence becomes

r^{n + 1} = (1 - \frac{3}{2} q) r^{n} + \frac{1}{2} q r^{n - 1},

(2)

so its amplification polynomial of a multilevel finite difference scheme is

p (r) = r^{2} - (1 - \frac{3}{2} q) r - \frac{1}{2} q .

(3)

Its roots are real. Moreover,

p (1) = q, p (- 1) = 2 (1 - q), \frac{r _{1} + r _{2}}{2} = \frac{1}{2} - \frac{3}{4} q .

(4)

The root-location criterion in the question therefore puts both roots in

[- 1, 1]

exactly when

0 \leq q \leq 1

. Fourier stability analysis for every wavenumber is equivalent to

4 μ \leq 1

, so

the scheme is stable if and only if μ \leq \frac{1}{4} .

(5)

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Put

k_{a} = k

ahead of the front and

k_{b} = k + Δ k

behind. With viscosity

μ

, Darcy's law gives the planar pressure gradients

p_{a 0}^{'} = - μU / k_{a}

and

p_{b 0}^{'} = - μU / k_{b}

. Write the displacement as

η e^{σ t + i α y}

with

α > 0

, and take the unperturbed interface to be

x = 0

in its translating frame.

The liquid flux is divergence-free away from the interface, so the pressure perturbations are harmonic. The decaying normal modes are

δ p_{b} = B e^{αx} e^{σ t + i α y}, δ p_{a} = A e^{- αx} e^{σ t + i α y} .

(1)

Expanding pressure continuity at the displaced interface and matching normal Darcy flux gives

B - A = μU (\frac{1}{k _{b}} - \frac{1}{k _{a}}) η = - \frac{μU Δ k}{k _{a} k _{b}} η, - k_{b} B = k_{a} A .

(2)

Hence

B = - \frac{μU Δ k}{k _{b} ( k _{a} + k _{b} )} η, A = \frac{μU Δ k}{k _{a} ( k _{a} + k _{b} )} η .

(3)

The common normal-flux perturbation is

δ u_{n} = - k_{b} α B / μ = αU Δ k η / (k_{a} + k_{b})

. The local enthalpy balance converts it into a front-speed perturbation,

σ η = δ u_{n} / (1 + S Δ ϕ)

. Thus the melting-front instability due to permeability contrast has

σ = \frac{αU}{1 + S Δ ϕ} \frac{Δ k}{2 k + Δ k} .

(4)

For a signed transverse wavenumber, replace

α

∣ α ∣

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Melting raises permeability of a porous medium. An advancing protrusion therefore offers a less resistive path to hot liquid, attracting larger Darcy flux and receiving more heat. The resulting faster melting reinforces the protrusion: this is a reactive infiltration instability. A permeability decrease would reverse this feedback, while zero contrast gives no growth in this ideal model.

For positive contrast, the growth rate is proportional to transverse wavenumber, so the model has no finite fastest-growing wavelength. The latent-to-sensible heat ratio slows the front but supplies no short-wave cutoff. Finite thermal transport and a resolved melting zone can introduce a length scale; pore geometry eventually limits the continuum description. If interface curvature changes the equilibrium melting temperature through the Gibbs--Thomson relation, that can also oppose short-wave corrugations. The modified dispersion relation must include such effects before a preferred wavelength can be calculated; a length scale alone does not guarantee that every added mechanism selects one.

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The linear dispersion relation for a mode

e^{i (k x - ω t)}

5 ω^{2} = k^{4} + 4

. The proposed modes share phase velocity

c

, so

5 c^{2} k_{j}^{2} = k_{j}^{4} + 4

. A quadratic wave interaction generates the second harmonic and the difference harmonic. With only two positive wavenumbers, phase matching for a quadratic wave interaction requires

k_{2} = 2 k_{1}

. Equating their phase velocities gives

k_{1}^{2} + \frac{4}{k _{1}^{2}} = 4 k_{1}^{2} + \frac{1}{k _{1}^{2}},

(1)

and therefore

k_{1} = 1, k_{2} = 2, c = 1.

(2)

This is a two-to-one resonance of dispersive waves. Its amplitude changes accumulate on

t = O (ϵ^{- 1})

. Choose the slow time

T = ϵ t /5

; this harmless constant rescaling makes the later amplitude formulas simple. With

ξ = x - t

, the leading real field is

A_{0} e^{i ξ} + B_{0} e^{2 i ξ} + \overline{A}_{0} e^{- i ξ} + \overline{B}_{0} e^{- 2 i ξ}

For the fundamental, the order-

ϵ

cross derivative in

5 ψ_{tt}

- 2 i (A_{0})_{T} e^{i ξ}

, while the fundamental coefficient in

ψ_{0} ψ_{0 x}

i \overline{A}_{0} B_{0}

. For the second harmonic the corresponding terms are

- 4 i (B_{0})_{T} e^{2 i ξ}

and

i A_{0}^{2}

. The solvability condition in the method of multiple scales removes the resonant forcing and gives

(A_{0})_{T} = - \frac{1}{2} \overline{A}_{0} B_{0}, (B_{0})_{T} = - \frac{1}{4} A_{0}^{2} .

(3)

The nonresonant third and fourth harmonics enter the correction. They do not change these leading amplitude equations.

Write

A_{0} = R e^{i θ}

B_{0} = P e^{i ϕ}

and

δ = ϕ - 2 θ

. Taking real and imaginary parts gives the explosive two-to-one amplitude system in polar form:

R_{T} = - \frac{RP}{2} cos δ, θ_{T} = - \frac{P}{2} sin δ, P_{T} = - \frac{R ^{2}}{4} cos δ, ϕ_{T} = \frac{R ^{2}}{4 P} sin δ .

(4)

The polar phases are used where their corresponding amplitudes are nonzero. These equations imply

(R^{2} - 2 P^{2})_{T} = 0, (R^{2} P sin δ)_{T} = 0.

(5)

For the second identity, differentiate using

δ_{T} = (R^{2} / (4 P) + P) sin δ

: the terms proportional to

sin δ cos δ

cancel. Therefore

R^{2} θ_{T} = - \frac{1}{2} R^{2} P sin δ, P^{2} ϕ_{T} = \frac{1}{4} R^{2} P sin δ

(6)

are first integrals, and in particular

(R^{2} θ_{T})_{T} = 0, (P^{2} ϕ_{T})_{T} = 0.

(7)

For constant phases with

ϕ = 2 θ + π

, these real equations reduce to

R_{T} = RP /2

and

P_{T} = R^{2} /4

. The initial data give

R^{2} - 2 P^{2} = 8

, so the Riccati equation for

P

2 P_{T} = P^{2} + 4

. Integrating and using

P (0) = 2

gives

P (T) = 2 tan (T + π /4), R (T) = 22 sec (T + π /4), 0 \leq T < π /4.

(8)

The tangent and secant function have a pole at

T = π /4

, corresponding to

t = 5 π / (4 ϵ)

. Both modes grow through their locked resonant interaction. This is formal finite-time blowup of the reduced amplitude equations; the weakly nonlinear expansion loses validity as the amplitudes become large, so it cannot establish a singularity of the full partial differential equation.

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Put

D = \partial_{t} + U \partial_{x}

, write

p

for perturbation pressure divided by

ρ_{0}

, and set

b = - g ρ^{'} / ρ_{0}

. The linearized Boussinesq approximation, incompressibility and material derivative give

D u + U^{'} w = - p_{x}, D w = - p_{z} + b, D b + N^{2} w = 0, u_{x} + w_{z} = 0.

(1)

Taking the curl eliminates pressure; differentiating the resulting vorticity equation in

x

and using incompressibility gives

D \nabla^{2} w - U^{''} w_{x} = b_{xx}

. A further application of

D

therefore yields

D^{2} \nabla^{2} w - U^{''} D \partial_{x} w + N^{2} \partial_{x}^{2} w = 0.

(2)

For a stationary nonzero horizontal Fourier mode,

D = U \partial_{x}

, so cancellation of

\partial_{x}^{2}

gives the stationary Taylor–Goldstein equation

(\nabla^{2} + \frac{N ^{2}}{U ^{2}} - \frac{U ^{''}}{U}) w = 0.

(3)

This division requires

U \neq = 0

on the interval considered: a zero of

U

is a critical level of an internal gravity wave. The mean profiles must be sufficiently smooth for the displayed derivatives, with background hydrostatic pressure and stable density stratification,

N^{2} > 0

, for propagating internal gravity waves. The horizontally uniform component is excluded from the cancellation. For a horizontal wavenumber

k

, local vertical propagation additionally requires

N^{2} / U^{2} - U^{''} / U > k^{2}

; this is distinct from the smoothness restrictions. If stability of the background against other disturbances is needed, the Miles–Howard theorem supplies the sufficient condition

N^{2} \geq (U^{'})^{2} /4

, rather than a necessary condition for deriving the equation.

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Take

U_{0}, N_{0}, k > 0

and assume an initially propagating mode,

k < N_{0} / U_{0}

. Put

s = N_{0} / U_{0}

and

δ (z) = 1 + γ z

. Neglecting

U^{''} / U

as stipulated, the Scorer parameter and stationary internal-wave WKB solution are

m (z)^{2} = \frac{s ^{2}}{δ ( z ) ^{2}} - k^{2}, w ≃ C m^{- 1/2} exp (ik x + i \int_{0}^{z} m (ζ) d ζ) .

(1)

The positive root

m > 0

selects upward group velocity for the negative intrinsic frequency branch

ω = - k U

. The WKB approximation needs a slowly varying background relative to the local vertical wavelength, in particular

∣ m^{'} ∣ ≪ m^{2}

and

∣ m^{''} ∣ ≪ ∣ m ∣^{3}

, away from a turning level or singular mean profile.

The local dispersion relation is

ω = k U - N k / (k^{2} + m^{2})^{1/2} = 0

. Its intrinsic and laboratory group velocities are, with

K = (k^{2} + m^{2})^{1/2}

c_{g, int} = \frac{N}{K ^{3}} (- m^{2}, km), c_{g} = \frac{U}{K ^{2}} (k^{2}, km) .

(2)

Thus intrinsic energy propagation is upstream and upward, while laboratory energy propagation is downstream and upward, parallel to

(k, m)

. Constant-phase lines of an internal gravity wave have slope

d z / d x = - k / m

and normal spacing

2 π / K

; the horizontal spacing remains

2 π / k

γ < 0

m

grows with height. The wavevector and laboratory energy ray become more vertical; phase lines become flatter and closer together. For finite nonzero

N

, the height

z_{c} = - 1/ γ

(3)

is a critical level of an internal gravity wave:

U \to 0

and the vertical wavenumber diverges. The laboratory group velocity tends to zero. Importantly,

∣ m^{'} ∣/ m^{2} \to ∣ γ ∣/ s

, so the formal WKB condition need not deteriorate before this level if that ratio is small. Nevertheless the stationary equation is singular at the level, the phase has no finite limit, and the horizontal perturbation velocity grows as

m^{1/2}

; the inviscid linear wave cannot be continued uniformly through it without additional physics.

γ > 0

m

decreases to zero at

z_{t} = \frac{s / k - 1}{γ} .

(4)

The wavevector and laboratory ray become horizontal, the phase lines become vertical, and the normal spacing increases. Above

z_{t}

m^{2} < 0

and the disturbance is an evanescent wave. Here

∣ m^{'} ∣/ m^{2}

diverges, so the WKB approximation fails in a turning region and must be replaced by a local connection solution.

Figure 1.
Rays of a stationary wave pattern
. Original WKB phase contours for decreasing and increasing $U / N$ . Red curves and arrows show laboratory energy rays; blue arrows show the local wavevector. The dashed levels mark the critical level and turning level, where a propagating WKB description cannot be continued unchanged.

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Use a common factor

e^{i (k x - ω t)}

, with

k = k_{0} > 0

and

ω = ω_{0}

. For the upward propagating branches with

0 < ω_{2} = ω - k U_{2} < N_{2}

, define

sin θ_{1} = \frac{ω}{N _{1}}, sin θ_{2} = \frac{ω _{2}}{N _{2}}, m_{j} = - k cot θ_{j}, 0 < θ_{j} < \frac{π}{2} .

(1)

These are the inclinations of upward-sloping constant-phase lines of an internal gravity wave to the horizontal. Write the incident, reflected and transmitted complex velocity amplitudes as

A_{0}, A_{r}, A_{t}

. Incompressibility gives

(u_{i}, w_{i}) = A_{0} (cos θ_{1}, sin θ_{1})

(u_{r}, w_{r}) = A_{r} (cos θ_{1}, - sin θ_{1})

, and

(u_{t}, w_{t}) = A_{t} (cos θ_{2}, sin θ_{2})

The jump conditions for stratified inviscid shear flow require a common interface displacement, not a common vertical velocity:

w = - i ω η

on each side. Since the background mass density is continuous, pressure is continuous without a hydrostatic jump. The horizontal momentum equation gives

p / ρ_{0} = (ω / k) u

. Hence

\frac{s i n θ _{1}}{ω} (A_{0} - A_{r}) = \frac{s i n θ _{2}}{ω _{2}} A_{t}, ω cos θ_{1} (A_{0} + A_{r}) = ω_{2} cos θ_{2} A_{t} .

(2)

These two matching equations determine internal-wave transmission across a velocity jump:

\frac{A _{t}}{A _{0}} = \frac{2}{r cos θ _{2} / cos θ _{1} + r ^{- 1} sin θ _{2} / sin θ _{1}}, \frac{A _{r}}{A _{0}} = \frac{r cos θ _{2} / cos θ _{1} - r ^{- 1} sin θ _{2} / sin θ _{1}}{r cos θ _{2} / cos θ _{1} + r ^{- 1} sin θ _{2} / sin θ _{1}}, r = \frac{ω _{2}}{ω} .

(3)

The amplitude formula printed in the PDF is inconsistent with these material-interface matching conditions. In particular, identical layers have

A_{t} / A_{0} = 1

and

A_{r} / A_{0} = 0

, whereas its expression labelled “reflected” equals one. The displayed results above distinguish reflection from transmission and retain the intrinsic-frequency factors required by the kinematic boundary condition and pressure balance.

For equal buoyancy frequencies and the specified opposing current,

θ_{1} = π /6

θ_{2} = arcsin (3/4)

, and

\frac{A _{t}}{A _{0}} = \frac{8}{4 + 21}, \frac{A _{r}}{A _{0}} = \frac{21 - 4}{21 + 4} .

(4)

The incident phase lines have slope

1/ 3

, the reflected lines slope

- 1/ 3

, and the transmitted lines slope

3/ 7

: they steepen above the interface. For the other specified current,

ω_{2} = 0

. The transmitted vertical wavenumber then diverges and there is no regular propagating upper-layer wave of that frequency: this is the critical level of an internal gravity wave limit. Approaching it from

ω_{2} > 0

gives

A_{r} / A_{0} \to - 1

and vanishing transmitted vertical energy flux; setting the intrinsic frequency to zero directly is outside the regular matching calculation.

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Write

x

for a comoving coordinate,

u

for the peculiar velocity,

H = \overset{a}{˙} / a

for the Hubble parameter, and

δ = δ ρ / \overset{ρ}{ˉ}

for the density contrast. For a homogeneous self-gravitating fluid, the linearized continuity equation, Euler equations for an inviscid fluid, and cosmological Poisson equation are

\dot{δ} = - \frac{1}{a} \nabla \cdot u, \dot{u} + H u = - \frac{c _{s}^{2}}{a} \nabla δ - \frac{1}{a} \nabla ϕ, \nabla^{2} ϕ = 4 π G a^{2} \overset{ρ}{ˉ} δ .

(1)

Here

c_{s}

is the physical adiabatic sound speed, gradients refer to comoving coordinates, and the homogeneous background force has already been subtracted. Differentiating the first equation and eliminating

u

and

ϕ

gives

\ddot{δ} + 2 H \dot{δ} - \frac{c _{s}^{2}}{a ^{2}} \nabla^{2} δ - 4 π G \overset{ρ}{ˉ} δ = 0.

(2)

A Fourier mode with comoving wavenumber

k

therefore obeys

\ddot{δ}_{k} + 2 H \dot{δ}_{k} + (\frac{c _{s}^{2} k ^{2}}{a ^{2}} - 4 π G \overset{ρ}{ˉ}) δ_{k} = 0.

(3)

The Jeans wavenumber occurs where the coefficient in parentheses vanishes. Since a comoving wavelength is

2 π / k

, the result is

k_{J} = \frac{a 4 π G ρ ˉ}{c _{s}}, λ_{J}^{com} = \frac{c _{s}}{a} \frac{π}{G ρ ˉ} .

(4)

Below this comoving Jeans length, pressure restores a displaced fluid element: there are acoustic waves rather than growing Jeans instability. In a static background they oscillate; expansion changes their amplitude and frequency, and dissipation can damp them. Pressure support alone does not erase them. In the pre-recombination photon-baryon fluid, Silk damping supplies a separate erasure mechanism.

For collisionless matter, replace the sound speed by a characteristic velocity dispersion. For an isotropic Maxwell-Boltzmann velocity distribution, defining

σ^{2} = ⟨ u_{x}^{2} ⟩ = ⟨ ∣ u ∣^{2} ⟩ /3

, the static collisionless marginal-stability calculation gives

λ_{J, collisionless}^{com} = \frac{σ}{a} \frac{π}{G ρ ˉ} = \frac{v _{rms}}{a} \frac{π}{3 G ρ ˉ} .

(5)

The coefficient depends on the distribution and the convention for velocity dispersion. Small-scale suppression is now collisionless free streaming and phase mixing, rather than collisional acoustic waves. An instantaneous collisionless Jeans length must be distinguished from the accumulated comoving distance

\int v d t / a

traveled since decoupling.

For the requested thermal histories, an Einstein-de Sitter universe describes the matter-era limit: it cannot literally also have a radiation-dominated epoch. Interpret the question as a spatially flat universe with negligible cosmological constant, passing from radiation domination,

a \propto t^{1/2}

, to matter domination,

a \propto t^{2/3}

. For the usual schematic dynamical Jeans scale use

\overset{ρ}{ˉ}_{tot}

in the estimate, so

λ_{dyn}^{com} \propto {c_{s} a, c_{s} a^{1/2}, t ≪ t_{eq}, \overset{ρ}{ˉ}_{tot} \propto a^{- 4}, t ≫ t_{eq}, \overset{ρ}{ˉ}_{tot} \propto a^{- 3} .

(6)

Near the particle horizon a relativistic perturbation calculation replaces the Newtonian derivation; the scaling estimate still identifies the relevant sound-crossing or streaming scale.

Before cosmological recombination, adiabatic initial conditions keep the entropy per baryon fixed in the tightly coupled photon-baryon fluid. With

R_{b} = 3 \overset{ρ}{ˉ}_{b} / (4 \overset{ρ}{ˉ}_{γ}) \propto a

, its photon-baryon sound speed is

c_{s}^{2} = \frac{c ^{2}}{3 ( 1 + R _{b} )} .

(7)

For example, this follows from

δ ρ_{γ} / \overset{ρ}{ˉ}_{γ} = (4/3) δ ρ_{b} / \overset{ρ}{ˉ}_{b}

and

δ P = c^{2} δ ρ_{γ} /3

. In the radiation-dominated limit

R_{b} ≪ 1

c_{s} ≃ c / 3

; in the strongly baryon-loaded matter-era limit

R_{b} ≫ 1

c_{s} \propto a^{- 1/2}

. Thus the baryon Jeans length across recombination has the asymptotic history

λ_{J, b}^{com} \propto ⎩ ⎨ ⎧ a \propto t^{1/2}, approximately constant, a^{- 1/2} \propto t^{- 1/3}, t ≪ t_{eq}, t_{eq} ≪ t < t_{rec}, t > t_{rec} with adiabatic gas cooling .

(8)

The transition to the plateau is smooth: retaining

1 + R_{b}

gives

λ_{J, b}^{com} \propto a^{1/2} / 1 + R_{b}

in matter domination. If baryon loading is still small, this intermediate segment instead rises as

a^{1/2}

until loading becomes important.

At photon decoupling, radiation ceases to provide pressure support to the baryons, so the sound speed drops from the coupled-fluid value to that of a nonrelativistic ideal gas,

c_{s, b}^{2} = γ k_{B} T_{b} / (μ m_{p})

, with

γ = 5/3

. The drop in comoving Jeans length at fixed density is the same ratio of sound speeds, typically several orders of magnitude. Subsequently an isentropic process gives

T_{b} \propto \overset{ρ}{ˉ}_{b}^{γ - 1} \propto a^{- 2}

, so

c_{s, b} \propto a^{- 1}

and

λ_{J, b}^{com} \propto a^{- 1/2}

. Residual energy exchange through Compton scattering can initially keep

T_{b} \propto a^{- 1}

, producing a short approximately flat gas segment before the decreasing segment. The idealized sketch assumes immediate adiabatic cooling:

log(comoving baryon Jeans scale)
  ^
  |                   __________________
  |                 /                   |
  |               /                     |  loss of photon pressure
  |             /                       |
  |           /                         +---\
  |         /                                \
  +-------------------|-----------------|----------> log(time)
                    t_eq              t_rec
        slope 1/2          slope ~0        slope -1/3

For the cold-dark-matter kinetic Jeans scale, take the stated toy history to mean that collisions maintain a particle temperature equal to the photon temperature until kinetic decoupling. This is an assumption about thermal coupling; a negligible collision rate after decoupling makes the subsequent treatment collisionless. Assume the ordering

t_{NR} < t_{dec} < t_{eq} < t_{rec}

. While relativistic, the characteristic particle speed is constant; while nonrelativistic but thermally coupled,

σ \propto T \propto a^{- 1/2}

; after kinetic decoupling, particle momentum redshifts as

a^{- 1}

and

σ \propto a^{- 1}

. Combining these with the background dynamical density gives

λ_{dyn, CDM}^{com} \propto ⎩ ⎨ ⎧ a \propto t^{1/2}, a^{1/2} \propto t^{1/4}, constant, a^{- 1/2} \propto t^{- 1/3}, t < t_{NR}, t_{NR} < t < t_{dec}, t_{dec} < t < t_{eq}, t > t_{eq} .

(9)

There is no sudden change at cosmological recombination: cold dark matter has already decoupled. Its small initial velocity dispersion makes its suppression scale much smaller than the pre-recombination baryon scale.

log(comoving CDM dynamical Jeans scale)
  ^
  |                        _______________
  |                    ___/               \
  |                ___/                    \       no jump
  |              /                          \      at t_rec
  |            /                             \
  +-----------|------------|--------------|----|------> log(time)
            t_NR         t_dec          t_eq t_rec
   slope 1/2    slope 1/4       slope 0       slope -1/3

The density convention matters. If one instead defines a species' self-gravitating collisionless Jeans length using only its own nonrelativistic density

\overset{ρ}{ˉ}_{m} \propto a^{- 3}

, its thermally coupled segment is constant and its decoupled segment decreases as

a^{- 1/2}

even during radiation domination. The constant radiation-era segment in the second sketch is the usual instantaneous background dynamical streaming estimate, not an exact single-species self-gravity threshold. A complete multicomponent calculation uses the separate perturbation equations and their combined cosmological Poisson equation.

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 333 / 1 / i / Solution 2026-09-29

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The basic state has velocity field

U = (0, Λ x, 0)

(1)

and buoyancy

B = N^{2} z

. For disturbances independent of

y

, the linearized equations are

u_{t} - f v = - p_{x}, v_{t} + (f + Λ) u = 0, w_{t} = - p_{z} + b,

(2)

b_{t} + N^{2} w = 0, u_{x} + w_{z} = 0.

(3)

Substituting a plane wave proportional to

exp [i (k x + m z - ω t)]

and eliminating

p

v

, and

b

gives the dispersion relation

ω^{2} = \frac{N ^{2} k ^{2} + f ( f + Λ ) m ^{2}}{k ^{2} + m ^{2}} .

(4)

Thus the requested coefficients are

f^{2} = f (f + Λ), g = N^{2} .

(5)

An instability exists precisely when some wavenumber pair makes

ω^{2} < 0

. Since the Brunt–Väisälä frequency satisfies

N^{2} > 0

, this is possible exactly when

f (f + Λ) < 0.

(6)

The basic relative vorticity is

Λ e_{z}

, so its absolute vorticity is

(f + Λ) e_{z}

. Its Ertel potential vorticity is therefore

Q = (f + Λ) N^{2} .

(7)

The instability criterion can consequently be written as

f Q < 0

: the vertical absolute vorticity has the opposite sign to the planetary vorticity. This is inertial instability, approached most directly by disturbances with

∣ m / k ∣ ≫ 1

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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 15B / d / Solution Created 2026-09-24 Updated 2026-09-25

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In the far past,

∣ c k τ ∣ ≫ 1

, the term

2/ τ^{2}

is negligible and the rescaled mode is a quantum harmonic oscillator of angular frequency

c k

. The stated choice of its ground state is the Bunch-Davies vacuum.

At late times, after the mode is far outside the Hubble scale,

∣ c k τ ∣ ≪ 1

. Its variance becomes

P_{k} = \frac{ℏ H _{inf}^{2}}{2 c ^{3} k ^{3}} (1 + c^{2} k^{2} τ^{2}) ⟶ \frac{ℏ H _{inf}^{2}}{2 c ^{3} k ^{3}} .

(1)

Since

P_{k}

has dimensions of length cubed, the dimensionless power is proportional to

k^{3} P_{k}

. More precisely, with the standard isotropic Fourier convention,

Δ^{2} (k) = \frac{k ^{3} P _{k}}{2 π ^{2}} ⟶ \frac{ℏ H _{inf}^{2}}{4 π ^{2} c ^{3}},

(2)

which is independent of the wavenumber

k

. Thus equal logarithmic intervals in

k

carry equal late-time power. This is the sense in which inflation naturally generates a scale-invariant inflationary power spectrum.

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Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 345 / 1 / e / Solution 2026-09-28

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The incident internal-wave ray has equation

z = - r x

. Intersecting it with the bottom

z = - H_{0} + s x

gives

x_{1} = \frac{H _{0}}{r + s} .

(1)

Its distance to the slope is consequently

ℓ = \frac{x _{1}}{cos θ} = \frac{H _{0}}{sin θ + s cos θ} .

(2)

The subcritically reflected ray rises through the same vertical distance at angle

θ

, so its next free-surface reflection is at

x_{2} = 2 x_{1} = \frac{2 H _{0}}{r + s} .

(3)

On the first leg, viscous attenuation of an internal-wave beam multiplies the energy density by

e^{- 2 k ℓ / Re}

. The bottom reflection multiplies it by

γ^{2}

and changes the wavenumber to

γk

. Since the corresponding Reynolds number is

Re / γ^{2}

, attenuation on the second leg contributes

e^{- 2 γ^{3} k ℓ / Re}

. Ignoring boundary-layer enhancement as requested,

\overline{E}_{2} = γ^{2} \overline{E}_{0} exp [- \frac{2 k ℓ}{Re} (1 + γ^{3})] .

(4)

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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 2 / f / Solution 2026-09-28

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Take the Fourier transform in

x

, with convention

v (k_{x}) = \int v (x) e^{- i k_{x} x} d x

. The differentiation rules

v^{''} = - k_{x}^{2} v

and

x^{2} v = - v^{''}

turn the forced equation into

S^{2} k_{y}^{2} \frac{d ^{2} v}{d k _{x}^{2}} + [κ_{r}^{2} + v_{s}^{2} (k_{x}^{2} + k_{y}^{2})] v = i [(2Ω - S) k_{x} ψ - S k_{y}^{2} \frac{d ψ}{d k _{x}}] .

(1)

For the unforced equation,

d k_{x} / d t = S k_{y}

implies

\frac{d ^{2}}{d t ^{2}} = S^{2} k_{y}^{2} \frac{d ^{2}}{d k _{x}^{2}} .

(2)

It is therefore exactly the shearing-wave oscillator found in part c, now parametrized by radial wavenumber rather than time.

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 38C / d / Solution Created 2026-09-24 Updated 2026-09-25

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The linearized equations are homogeneous and have constant coefficients in

x

and

t

. Consequently, spatial Fourier modes evolve independently, and the superposition principle reconstructs arbitrary sufficiently regular initial disturbances from them. It is therefore enough to seek the normal modes

η (x, t) = Re {η_{0} e^{ik x + σ t}} .

(1)

Here the real

k

is the wavenumber,

σ

is the possibly complex temporal exponent, and taking the real part produces a physical displacement.

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 332 / 3 / iii / Solution 2026-09-28

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Write the base-state gradients as

G_{w} = \frac{T _{h} - T _{m}}{h}, G_{a} = \frac{T _{a} - T _{h}}{δ} .

(1)

The base heat flux and Stefan condition relations are

k_{w} G_{w} = k_{a} G_{a}, - k_{w} G_{w} = ρ_{i} L_{f} V .

(2)

Under the quasi-stationary approximation, and with advective heat transport neglected, each temperature perturbation satisfies the Laplace equation. The decaying normal-mode solutions are

T_{w} = A cosh (αx) + B sinh (αx), T_{a} = C e^{- α (x - h)} .

(3)

Keeping the displaced ice interface at

T_{m}

omits the curvature correction described by the Gibbs--Thomson relation.
The fixed melting temperature at the displaced ice interface gives

A = - η_{1} G_{w} .

(4)

Expanding temperature and conductive heat flux continuity at the displaced outer interface gives

A cosh (α h) + B sinh (α h) - C = η_{2} (G_{a} - G_{w}),

(5)

k_{w} [A sinh (α h) + B cosh (α h)] = - k_{a} C .

(6)

Put

r = k_{a} / k_{w}

. Since

G_{a} = G_{w} / r

, the temperature condition in the limits

r ≪ 1

and

α h ≪ 1

gives

C \sim - \frac{η _{2} G _{w}}{r} .

(7)

Although

r

is small, the product

r C

is therefore order one and cannot be discarded. The flux condition then gives

B = - r C - A α h + o (1) \sim η_{2} G_{w} .

(8)

This is why the stated asymptotic warning matters.

The perturbed Stefan condition at

x = 0

ρ_{i} L_{f} σ η_{1} = - k_{w} T_{w}^{'} (0) = - k_{w} α B .

(9)

Using

B \sim η_{2} G_{w}

and

- k_{w} G_{w} = ρ_{i} L_{f} V

yields

σ η_{1} = V α η_{2} .

(10)

Substitution of the interface-amplitude ratio from part i produces the thin-film icicle-ripple instability dispersion relation

\frac{hσ}{V} = \frac{α h}{1 - i Γ α ^{3} h} .

(11)

Set

b = Γ h α^{3}

. Rationalizing this complex number gives

σ = \frac{V α ( 1 + ib )}{1 + b ^{2}},

(12)

so its growth rate and imaginary part are

Re σ = \frac{V α}{1 + ( Γ h ) ^{2} α ^{6}}, Im σ = \frac{V α b}{1 + b ^{2}} .

(13)

Differentiating the growth rate with respect to the wavenumber shows that its positive stationary point satisfies

1 - 5 (Γ h)^{2} α_{m}^{6} = 0.

(14)

It is the unique global maximum, and hence

α_{m} = 5^{- 1/6} (Γ h)^{- 1/3} .

(15)

At this wavenumber

b = 1/ 5

. With the convention

e^{i α z + σ t}

, a constant phase travels with phase velocity

c_{p} = - \frac{Im σ}{α} = - \frac{Vb}{1 + b ^{2}} .

(16)

Therefore

c_{p} = - \frac{5}{6} V .

(17)

Because

z

increases downward, the negative sign means that the icicle ripples migrate upward with speed

5 V /6

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 333 / 3 / e / Solution 2026-09-28

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Multiply

w_{zz} + (l^{2} - k^{2}) w = 0

(1)

w

and integrate from the rigid bottom to an upper endpoint

z

at which

w w_{z} = 0

. The bottom term also vanishes because

w (0) = 0

. Integration by parts gives

- \int_{0}^{z} w_{z^{'}}^{2} d z^{'} + \int_{0}^{z} (l^{2} - k^{2}) w^{2} d z^{'} = 0.

(2)

Therefore the horizontal wavenumber has the Rayleigh quotient

k^{2} = \frac{\int _{0}^{z} ( l ^{2} w ^{2} - w _{z^{'}}^{2} ) d z ^{'}}{\int _{0}^{z} w ^{2} d z ^{'}} .

(3)

For a trapped wave the upper endpoint may be taken to infinity because the eigenfunction decays.

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 345 / 2 / b / ii / Solution 2026-09-28

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For a face of signed slope

s

, conservation of frequency and tangential wavenumber gives, on the branch relevant to an incident downward-right ray,

k_{r} = k \frac{1 + s μ}{1 - s μ}, μ = \frac{∣ m ∣}{∣ k ∣} = cot θ .

(1)

s μ = 1

, the reflected wavenumber diverges and the reflected group velocity becomes tangent to the face. On a supercritical face the denominator changes sign: horizontal propagation reverses and rays from the two faces are directed towards a sawtooth corner. Successive reflections therefore focus energy and shorten the wavelength.

The inviscid ray pattern cannot persist indefinitely. Near-critical focusing amplifies gradients until kinematic viscosity, mass diffusivity, nonlinear wave steepening, and wave breaking matter; a real corner is also rounded on some finite scale. These effects replace the singular ray construction by dissipative boundary layers, mixing, and a finite-width reflected beam.

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