Rayleigh equation for inviscid shear flow

For a parallel inviscid base flow $U(z)$ and disturbance streamfunction $\varphi (z)e^{i\alpha (x-ct)}$, the Rayleigh equation is

For $U=-1$ below $z=-1$, $U=z$ between $z=-1$ and $z=1$, and $U=1$ above $z=1$, decaying normal modes solve ${\varphi }^{\prime \prime }-k^2\varphi =0$ in each region. Pressure matching at a piecewise-linear shear interface gives continuity of $\varphi$ and $(U-c){\varphi }^{\prime }-U^{\prime }\varphi$. The two corner conditions yield the displayed dispersion relation. At $k=1/2$ the wave speeds are $c=\pm i/e$, so one mode grows exponentially with rate $1/(2e)$.

An axisymmetric Stokes streamfunction disturbance $\varphi (r)e^{ik(z-ct)}$ of an axial basic flow obeys the displayed ordinary differential equation. Regularity requires $\varphi =O(r^2)$ at the axis, and an impermeable wall requires $\varphi (a)=0$ for nonzero $k$. Multiplication by ${\varphi }^{\ast }/r$ and integration by parts shows that a growing real-wavenumber mode requires $(U^{\prime }/r{)}^{\prime }$ to change sign; if it vanishes identically, the positive energy integral rules out such a mode. The condition concerns this cylindrical vorticity gradient rather than $U^{\prime \prime }$ alone.

When $U^{\prime }/r$ is constant and $U\ne c$, the axisymmetric inviscid pipe stability equation reduces to ${\varphi }^{\prime \prime }-{\varphi }^{\prime }/r-k^2\varphi =0$. Axis regularity selects $r{I}_1(kr)$ and impermeability gives $I_1(ka)=0$. These imaginary-wavenumber disturbances are spatially evanescent potential fields, not bounded whole-pipe temporal normal modes; their radial equation supplies no eigenvalue condition on $c$.

Decaying normal modes of this continuous piecewise-linear profile satisfy ${\varphi }^{\prime \prime }-k^2\varphi =0$ between the corners. Pressure matching at a piecewise-linear shear interface yields an odd varicose mode of a planar jet with real speed $c=(1-e^{-2k})/(2k)$ and an even sinuous mode of a planar jet whose speeds satisfyIts reduced discriminant factors as $(2k-3-e^{-2k}-4e^{-k})(2k-3-e^{-2k}+4e^{-k})$. The second factor is positive for $k>0$, while the first is strictly increasing and crosses zero once. Thus exactly the long-wave even branch is exponentially unstable, for $0<k<k_c$, where $2k_c-3-e^{-2k_c}-4e^{-k_c}=0$.

For a normal mode of an inviscid parallel flow, the perturbation pressure is $p=-(U-c){\varphi }^{\prime }+U^{\prime }\varphi$. At an interface where $U$ is continuous but $U^{\prime }$ jumps, continuity of normal velocity and pressure gives continuity of $\varphi$ and the displayed flux. Consequently $(U-c)[{\varphi }^{\prime }]=[U^{\prime }]\varphi$. A sinuous mode of a planar jet may therefore have a cusp at a symmetric velocity-profile corner: even parity alone does not force its one-sided derivative to vanish.

For a smooth inviscid parallel shear profile with an exponentially growing mode, let $U_s$ be its velocity at an inflection point. The real and imaginary parts of the integrated Rayleigh equation for inviscid shear flow imply $\int U^{\prime \prime }(U-U_s)|\psi {|}^2/|U-c{|}^2=-\int (|{\psi }^{\prime }{|}^2+k^2|\psi {|}^2)<0$. Consequently $U^{\prime \prime }(U-U_s)<0$ somewhere in the channel. This strengthens Rayleigh's inflection-point theorem but is still only a necessary condition, not a sufficient instability criterion.

For inviscid plane Couette flow $U=z$ between impermeable walls at $z=\pm 1$, the nonzero real streamwise wavenumber $\alpha$ gives no nontrivial smooth discrete vertical-velocity modes. Instead, the vorticity variable $q=(D^2-{\alpha }^2)w$ obeys $({\partial }_t+i\alpha z)q=0$. Localized vorticity sheets give generalized eigenfunctions with real frequencies ${\omega }_{\xi }=\alpha \xi$. Their closure forms a neutral continuous spectrum.

For constant shear $U=\lambda y$, the wall-normal Laplacian of velocity is advected with phase $e^{-i\alpha \lambda yt}$. Inverting $D^2-k^2$, $k^2={\alpha }^2+{\beta }^2$, with its Dirichlet Green function gives the displayed initial-value solution. Impermeable inviscid walls impose zero normal velocity, not zero tangential velocity or no-slip wall vorticity.

In unbounded Couette flow with velocity $(y,0)$, linear vorticity is transported exactly as $\omega (x,y,t)={\omega }_0(x-yt,y)$. For a nonzero streamwise Fourier mode $\alpha$ and sufficiently localized smooth initial data, inversion by the one-dimensional modified Helmholtz Green function gives the displayed pointwise decay of the streamfunction and velocity, although vorticity does not decay in magnitude. The mechanism is phase mixing: shearing produces increasingly rapid transverse oscillations, whose elliptic velocity reconstruction suppresses them. Smoothness and decay of the initial streamfunction without derivative control do not alone imply these algebraic rates.

Let $G_a$ be the Dirichlet Green function of $D^2-a^2$ on $[-1,1]$, with $a=|\alpha |$:It is continuous and has derivative jump one, so $(D^2-a^2)G_a=\delta (z-\xi )$. The Dirac delta multiplication identity gives $(z-\xi )\delta (z-\xi )=0$, proving it is a generalized eigenfunction of the inviscid Couette continuous spectrum. Integrating $G_a(z,\xi )q_0(\xi )e^{-i\alpha \xi t}$ reconstructs the evolving vertical velocity from its initial vorticity.

Across a jump of $U^{\prime }$ in a constant-density parallel flow with continuous $U$, the normal mode vertical velocity and pressure are continuous. The linearized momentum equation gives $\hat{p}=\rho [(U-c){\hat{w}}^{\prime }-U^{\prime }\hat{w}]/(ik)$, yielding the displayed conditions. Equivalently, away from $U=c$, $(U-c)[{\hat{w}}^{\prime }]=[U^{\prime }]\hat{w}$. Derivative continuity is generally incorrect when the background vorticity jumps. This is the constant-density, continuous-velocity specialization of jump conditions for stratified inviscid shear flow.

An inviscid parallel shear flow has base velocity $\mathbf{u}=U(y){\mathbf{e}}_x$. Two-dimensional normal-mode perturbations satisfy the Rayleigh equation for inviscid shear flow.

A uniform-density inviscid shear flow with $U=U_{0}z/d$ for $|z|<d$ and $U=\pm U_0$ outside has two vorticity jumps but no walls. Decaying normal modes satisfy $w^{\prime \prime }-k^{2}w=0$ in each region. Continuity of $w$ and pressure matching at a piecewise-linear shear interface gives a two-by-two determinant whose diagonal entries are $1-2kd\pm 2kdc/U_0$ and off-diagonal entries $e^{-2kd}$. Its vanishing gives the displayed dispersion relation. The Kelvin-Helmholtz instability band is $0<kd<{\alpha }_s$, where $2{\alpha }_s-1=e^{-2{\alpha }_s}$ has its unique positive root above $1/2$, approximately $0.63923227$. Short waves decouple the two edges and remain neutral; long waves approach the vortex sheet instability with $c=\pm i{U}_0$.

This three-layer shear flow has $U=\Delta Uz/h$ and density ${\rho }_0$ between $z=\pm h/2$. Above and below, the velocity is the corresponding constant $\pm \Delta U/2$, and the mass density is ${\rho }_0\mp \Delta \rho /2$. The density drops and vorticity jumps coincide at both interfaces. Its hydrodynamic stability involves coupled gravity-vorticity interface waves, unlike a profile with uniform shear extending through all three layers.

For the bounded-shear three-layer profile with density contrasts minus one, zero and plus one, the localized-mode matching determinant is $[a(1-c{)}^2-(1-c)-J][a(1+c{)}^2-(1+c)-J]-b^2(1-c^2{)}^2=0$, where $a=\alpha [1+\coth (2\alpha )]$ and $b=\alpha \mathrm{csch}(2\alpha )$. Dividing by $a^2-b^2=2\alpha a$ gives a quartic in $c$ and a quadratic in $c^2$. Its constant term is $\{[2\alpha -(1+J){]}^2-e^{-4\alpha }(1+J{)}^2\}/(4{\alpha }^2)$, whose negative sign proves the unstable band of a three-layer stratified shear flow.

For the three-layer stratified piecewise-linear shear flow, set $\alpha =kh/2$. The dispersion relation is quadratic in squared nondimensional phase velocity. Its constant term is $[2\alpha -(1+J)(1+e^{-2\alpha })][2\alpha -(1+J)(1-e^{-2\alpha })]/(4{\alpha }^2)$. Throughout the displayed band this is negative, so one squared speed is negative and gives an exponentially growing normal mode.

The Hazel model is a smooth stratified shear flow whose base velocity has a hyperbolic-tangent profile and whose squared buoyancy frequency has a hyperbolic-secant-squared profile. The equal-width nondimensional example is $U=\tanh z$ and $N^2=J{\mathrm{sech}}^{2}z$. For $J\ge 0$, its gradient Richardson number is $J{\cosh }^{2}z$, with minimum $J$.

The zero-phase velocity ansatz $w=(\mathrm{sech}z{)}^k(\tanh z{)}^{1-k}$ solves the Taylor–Goldstein equation away from its critical level of an internal gravity wave exactly when $J=k(1-k)$. Let $y=\tanh z$ and $a=1-k$. Then $w^{\prime }/w=a/y-y$ and $w^{\prime \prime }/w=a(a-1)/y^2-a-1+2y^2$, so the differential-equation residual divided by $w$ is $[J-k(1-k)](y^{-2}-1)$. The curve has maximum $1/4$ at $k=1/2$. The endpoint $k=0$ does not decay at infinity; fractional powers at the critical level require a branch and regularity qualification.

For $0<k<1$, a neutral mode of the equal-width Hazel model behaves as $w\sim z^{1-k}$ near $z=0$. Its derivative diverges as $z^{-k}$, so it is not a classical continuously differentiable mode across the critical level of an internal gravity wave. The horizontal velocity is proportional to $w^{\prime }/k$; its local kinetic energy is finite only for $k<1/2$, because ${\int }_0^{ϵ}{z}^{-2k}dz$ converges precisely then. One possible neutral-mode convention is the boundary value from $U-c$ with $c_i\downarrow 0$, which fixes the phase of the power on the negative-$z$ side. The Miles–Howard theorem concerns growing modes with nonreal $c$ and does not rule out such singular neutral limits.

Let $\zeta =z/L\in [-1,1]$, $h=L_s/L\in (0,1)$, and normalize velocity by its maximum magnitude. The profile is $U=\zeta /h$ for $|\zeta |<h$ and $U=\mathrm{sgn}\zeta$ outside. Each region has exponential vertical-velocity modes; the two walls and vorticity-jump matching for an inviscid shear flow determine their coupling. With $\alpha =kL$, $q=\alpha h$, $X=\tanh q$, $Y=\tanh [\alpha (1-h)]$, $H=q(1+XY)$ and $J=q(X+Y)$, elimination gives $c^2=(H-Y)(J-XY)/(HJ)$, with $c$ normalized by the velocity scale.

The bounded piecewise-linear shear layer has exponentially growing modes exactly when $h<1/2$. Its squared phase speed tends to $2h-1$ as $kL\to 0$ and to $1$ as $kL\to \infty$. For $h<1/2$, sufficiently long nonzero waves are therefore unstable. Conversely $J-XY>0$, and for $h\ge 1/2$, $q\ge \alpha (1-h)>Y$, so $H-Y>0$ for every positive wavenumber and $c^2>0$. This proves the threshold without assuming monotonicity of the dispersion curve.

An inviscid parallel shear flow can possess an exponentially growing normal mode only if $U^{\prime \prime }$ changes sign somewhere in the flow.

If an inviscid shear-flow mode has complex wave speed $c$ and $U_{-}\le U\le U_{+}$, then

For a nonreal regular Rayleigh equation for inviscid shear flow mode vanishing at the walls, put $F=\hat{w}/(U-c)$. The equation becomes $[(U-c{)}^2{F}^{\prime }{]}^{\prime }-k^2(U-c{)}^{2}F=0$. Multiplication by $\overline{F}$ and integration by parts gives the displayed identity. With $Q=|F^{\prime }{|}^2+k^2|F{|}^2$, its imaginary and real parts give $\int UQ=c_r\int Q$ and $\int U^{2}Q=|c{|}^2\int Q$. Since $(U-U_{\min })(U_{\max }-U)\ge 0$, these imply Howard's semicircle theorem.