Fluid mechanics

Fluid mechanics studies the motion and forces of liquids and gases.

A pathline is a fluid particle's material trajectory through a time-dependent velocity field. Unlike a streamline, which follows the velocity field at one frozen time, a pathline solves the displayed ODE. The two curves coincide in a steady flow but generally differ in an unsteady one.

A control volume is a chosen region over which fluid mass conservation and momentum balance are integrated. It may be fixed or move with a prescribed velocity. Boundary fluxes must use velocity relative to the moving boundary. In Stokes flow, negligible inertia reduces the momentum balance to the sum of body forces and boundary tractions, making a carefully chosen control volume useful for finding drag without integrating directly over a curved body.

A Navier slip boundary condition relates fluid velocity relative to a stationary flat wall to the wall-normal tangential velocity gradient, $u_b={\ell }_s{u}_z$, with positive slip length ${\ell }_s$. For a Newtonian fluid, it is equivalent to linear resisting wall traction of magnitude $\mu u_b/{\ell }_s$. The normal points from the wall into the fluid; this fixes the sign convention.

The slip length is the distance obtained by extrapolating a locally linear tangential velocity profile back to zero velocity beneath a wall. A Navier slip boundary condition has $u_b={\ell }_s{\partial }_{n}u$. Zero slip length gives a no-slip boundary condition; large slip permits approximately plug-like motion when the surrounding geometry and forces allow it.

A fluid free boundary is an initially unknown material interface separating a fluid from its exterior. For a polytropic elliptical patch in a shearing sheet bounded by vacuum, it is the level set $p=\rho =Q=0$, and the velocity is tangent to it in a steady solution.

A material surface is transported with its fluid elements, including its boundary when open. Its oriented material surface element evolves by tangent transport. Magnetic flux freezing conserves the flux through such an open surface.

If $\mathbf{A}={\mathbf{t}}_a\times {\mathbf{t}}_{b}dadb$ and each tangent satisfies $D\mathbf{t}/Dt=(\nabla \mathbf{u})\mathbf{t}$, differentiation of the cross product gives $D\mathbf{A}/Dt=(\nabla \cdot \mathbf{u})\mathbf{A}-(\nabla \mathbf{u}{)}^T\mathbf{A}$. Combining this with ideal induction proves pointwise conservation of $\mathbf{B}\cdot \mathbf{A}$.

A material curve consists of the same fluid elements at all times. Differentiating their flow map gives $D\mathbf{\ell }/Dt=(\mathbf{\ell }\cdot \nabla )\mathbf{u}$ for its tangent. In ideal magnetohydrodynamics a magnetic-field tangent obeys this same transport law after division by mass density.

Scalar transport follows a material quantity through advection and diffusion equation terms, for example ${\Theta }_t+\mathbf{U}\cdot \nabla \Theta =\kappa \Delta \Theta$. A passive scalar does not affect the velocity field; an active scalar feeds back through forces such as buoyancy. With incompressible flow bounded by impermeable walls and homogeneous Neumann boundary conditions, the spatial mean is conserved.

An active scalar is a transported scalar field that feeds back on the velocity field, for example a density anomaly producing buoyancy. In an adjoint equations for Boussinesq scalar mixing calculation its dynamical coupling must also be transposed; a passive-scalar sensitivity equation generally omits that feedback.

A passive scalar is an advected and diffusing scalar field whose value has no dynamical effect on the transporting velocity field. Concentration of a dilute nonreacting dye is a common example. Its transport can depend on the flow while the flow equations remain independent of it.

The horizontally integrated plume concentration is the integral of a local concentration $C$ across the plume. It measures scalar amount per unit vertical length and unit span, and its vertical integral is total scalar amount per unit span. Unlike a local concentration, its dimensions include one horizontal length. It must be distinguished from horizontally averaged plume concentration when a plume's width varies with height.

For a plume of full width $w(z)$, its horizontally averaged concentration is $\overline{C}(z,t)=c(z,t)/w(z)$, where $c$ is horizontally integrated plume concentration. In a line plume with $w=\mu z$, division by $z$ generally changes the location of maximum concentration. A gradient closure for $c$ differs from one for $\overline{C}$: $w{\partial }_z\overline{C}=c_z-c/z$, so integrated diffusive flux $-Kw{\partial }_z\overline{C}$ equals $-K(c_z-c/z)$.

For a smooth incompressible direct trajectory with total velocity field $\mathbf{U}$ and active scalar $\Theta$, the negative-constraint Lagrangian function in constrained optimization convention givesThe transpose term is the formal adjoint of $\delta \mathbf{u}\cdot \nabla \mathbf{U}$; the two couplings transpose scalar transport by advection and buoyancy. For terminal cost $\int \Theta (T{)}^2$, the terminal data are ${\mathbf{u}}^{†}(T)=0$, ${\theta }^{†}(T)=2\Theta (T)$. They are integrated backward along the stored direct trajectory. No-slip boundary conditions and Dirichlet boundary conditions for the adjoint velocity and homogeneous Neumann boundary conditions for the adjoint scalar remove the spatial boundary terms.

Scalar variance measures spatial departure from the conserved mean. For zero mean, incompressible flow bounded by impermeable walls and homogeneous scalar Neumann boundary conditions, integration by parts gives $d\int {\Theta }^2/dt=-2\kappa \int |\nabla \Theta {|}^2$. Advection preserves the instantaneous quadratic integral but can sharpen gradients, allowing the diffusion equation to remove scalar variance faster.

A fluid flow is motion described by a velocity field $\mathbf{u}(\mathbf{x},t)$. The trajectories of fluid elements satisfy $d\mathbf{x}/dt=\mathbf{u}(\mathbf{x},t)$; the mass density and other fields obey their conservation laws and constitutive equations. This physical motion differs from a flow on a flow network.

A unidirectional flow has velocity parallel to a single fixed direction, for example $\mathbf{u}=u(y){\mathbf{e}}_x$. If the speed is independent of the coordinate along that direction, the convective acceleration $(\mathbf{u}\cdot \nabla )\mathbf{u}$ vanishes. The steady Navier-Stokes equation then reduces to a linear balance of the pressure gradient and viscosity, as in Couette flow and Poiseuille flow.

In steady flow, the velocity and other relevant flow fields are independent of time in the chosen reference frame. Streamlines then coincide with the paths of moving particles, while convective acceleration can still be nonzero.

An inviscid flow neglects viscosity in the fluid flow. Its momentum balance is described by the Euler equations for an inviscid fluid.

Sediment resuspension transfers deposited bed particles into a particle suspension. It can supply particle buoyancy flux to a particle-laden gravity current if erosion or upward mixing exceeds settling losses. Ambient fluid without particles dilutes concentration without supplying particle mass.

The equilibrium sediment flux is the transport supported locally by a prescribed bed shear stress. A common idealization is $q_{\mathrm{sat}}={\varphi }_b\chi (\tau -{\tau }_{\mathrm{th}}{)}_{+}^{\gamma }$. Linearization above the sediment entrainment threshold requires perturbations small compared with the positive excess stress.

The saturation length measures downstream relaxation of a transported grain population toward its equilibrium sediment transport flux. The elementary steady relaxation law is $L_{\mathrm{sat}}{q}_x=q_{\mathrm{sat}}-q$; it introduces a phase lag between forcing and actual transport.

The Exner equation is local sediment mass conservation expressed in terms of solid volume: accumulation raises the bed and flux divergence lowers it. Here ${\varphi }_b$ is the solid volume fraction of the packed bed and $q$ is solid volume flux per unit width.

The Shields parameter is $\Theta =\tau /[({\rho }_p-{\rho }_f)gd]$, comparing bed shear stress with a grain-scale submerged weight per area. Its critical value depends on the assumed drag coefficient, static friction, lift, and contact geometry.

The sediment entrainment threshold is the bed shear stress at which resting grains begin moving. An elementary force balance using spherical grains, tangential quadratic drag, and static friction gives ${\Theta }_{\mathrm{th},0}=4{\mu }_s/(3C_D)$ when the drag coefficient is defined relative to the bed shear velocity.

For locally bed-tangent drag force and an upslope angle $a$, a spherical-grain force balance gives ${\tau }_{\mathrm{th}}(a)={\tau }_{\mathrm{th},0}(\cos a+\sin a/{\mu }_s)$. The small-slope term is $\delta {\tau }_{\mathrm{th}}=({\tau }_{\mathrm{th},0}/{\mu }_s){\eta }_x$. A horizontal drag force instead changes the normal contact force and gives a different slope correction.

Combining the Exner equation, sediment transport saturation, bed shear response, and inclined-bed sediment threshold gives a competition between upstream forcing and downstream relaxation. With $a={\tau }_{0}A$, $b={\tau }_{0}B-{\tau }_{\mathrm{th},0}/{\mu }_s$, $C=\chi \gamma ({\tau }_0-{\tau }_{\mathrm{th},0}{)}^{\gamma -1}$, and positive wavenumber $k$, the growth rate is $\sigma =C{k}^2(b-ak{L}_{\mathrm{sat}})/(1+k^2{L}_{\mathrm{sat}}^2)$. Thus $b>0$ permits long-wave growth when $a>0$, while a finite saturation length stabilizes shorter waves.

For a small sinusoidal bedform, a fluid-dynamical closure can be written $\hat{\tau }={\tau }_{0}k(A+iB)\hat{\eta }$ for $k>0$. The real coefficient $A$ is in phase with the bed height and $B$ describes an upstream phase lead. Neither coefficient follows from the Exner equation or sediment relaxation alone; they require a flow calculation or measurement.

The momentum-flux pressure associated with a moving fluid. Some energy-density balances use $\rho v^2/2$ instead; the numerical convention must be stated when balancing magnetic pressure.

Turbulence is irregular fluid motion with interacting velocity fluctuations over a range of scales. It can mix a particle suspension and promote fluid entrainment into a turbulent plume.

Mixing by turbulence is represented approximately by a diffusive flux down a mean concentration gradient. Its coefficient is the eddy diffusivity. A constant-coefficient model gives a $D{\nabla }^2\sigma$ term and a smoothing rate $D{k}^2$ for a Fourier mode. Effective diffusion can suppress slow gravitational concentration even when microscopic diffusion is negligible.

For positive dense-fluid reduced gravity $G$ and upward coordinate $z$, an unstable gradient $G_z>0$ supplies buoyant work of order $d^2{G}_z$ over mixing length $d$. velocity $u\sim d\sqrt{G_z}$ and eddy diffusivity $K\sim ud$ follow. Order-one constants depend on the turbulence model. This closure is for an unstable gradient; its square root must not be applied to a stable negative gradient.

An upward volume flux $Q$ and a downward turbulent buoyancy transport $d^4(G_z{)}^{3/2}$ balance in a steady reactor. Matching $G=G_z=0$ below $-H$ gives $QG=d^4(G_z{)}^{3/2}$. Integrating yields the cubic profile above the front. If downward source buoyancy flux is $B_s$, then $H=3d^{8/3}{B}_s^{1/3}/Q$. The corresponding eddy diffusivity is $K=Q(z+H)/(3d^2)$. All these prefactors use the same area and mixing-length normalization; an actual cross-sectional area $S$ replaces the denominator in $K$ by $3S$.

In the area normalization $d^2$, the buoyancy-gradient mixing-length closure transports dense-fluid buoyancy downward with magnitude $\mathcal{F}=d^{2}K{G}_z=d^4(G_z{)}^{3/2}$. Its convergence in an upward coordinate is ${\mathcal{F}}_z$. Transport has units $L^4{T}^{-3}$ and convergence units $L^3{T}^{-3}$. Constant positive $G_z$ has nonzero transport and zero convergence; the two quantities cannot be identified.

An eddy diffusivity is a transport closure representing unresolved turbulence by a mean scalar flux proportional to a negative mean concentration gradient. A local estimate is $K\sim U\ell$, where $U$ and $\ell$ are turbulent speed and length scales. In a line plume with constant characteristic speed and width proportional to $z$, this gives $K\propto z$. Its coefficient is an independent closure parameter; fluid entrainment conservation alone does not determine it.

A circular turbulent round jet supplied with kinematic momentum flux $m_0$ but negligible buoyancy flux conserves $m=m_0$. A top-hat plume model and constant Batchelor entrainment give $dq/ds=2\alpha \sqrt{\pi m_0}$. From a point source with zero volume flux, $q=2\alpha \sqrt{\pi m_0}s$, radius $b=2\alpha s$, and axial velocity $V=m_0/q\propto s^{-1}$.

An acoustic wave is a propagating pressure and density disturbance in a material medium. Linear sound waves oscillate when pressure restores a displaced fluid element.

An acoustically rigid boundary has zero perturbation normal velocity. For a reflected acoustic plane wave, pressure amplitudes add in phase and the reflection coefficient is $+1$. It is the infinite-surface acoustic impedance limit. A pressure-release boundary instead fixes the pressure perturbation to zero and reflects with coefficient $-1$.

A stratified atmosphere can reflect low-frequency acoustic waves while permitting sufficiently high-frequency waves to escape. In a plane-parallel isothermal atmosphere, the angular acoustic cutoff is approximately $c_s/(2H_{\rho })$, where $H_{\rho }$ is the density scale height. Its detailed form depends on the atmospheric profile and on the wave-variable approximation.

A pressure-release boundary imposes zero acoustic pressure disturbance, $\tilde{p}=0$. The incident and reflected pressure amplitudes therefore obey $1+R=0$, so the pressure reflection coefficient is $R=-1$. It contrasts with a rigid inviscid boundary, where zero normal velocity gives $R=+1$. A much lower normal impedance on the far side approximates the pressure-release condition.

A velocity field assigns a velocity vector to every position and time in a moving continuum. Its spatial derivatives describe local deformation, rotation, and volume change.

The vertical velocity is the component of a fluid velocity field along the selected upward vertical direction. In incompressible flow, impermeable horizontal boundaries impose $w=0$. Eliminating pressure often yields a closed hydrodynamic stability equation for this component.

An axisymmetric flow is invariant under rotations about one fixed axis, so scalar fields and cylindrical-coordinate velocity components do not depend on the azimuthal angle.

Advection transports a quantity with a velocity field. For a scalar field $f$, its local advective rate is $\mathbf{u}\cdot \nabla f$, the spatial contribution to the material derivative. The same term transports the components of a vector field.

Convection transports properties such as energy or composition through bulk fluid motion. Thermal convection is driven by temperature-dependent buoyancy.

A reduced long-wave convection model can combine linear damping, destabilizing second derivatives, stabilizing fourth derivatives and gradient nonlinearities. In the displayed normalization the quadratic coefficient $s$ breaks Boussinesq symmetry, while the cubic gradient term dissipates the mean-square temperature. The linear dispersion relation is $\lambda (k)=-1+\mu k^2-k^4$, with critical wavenumber $k=1$ at $\mu =2$. Smoothness and spatial averaging hypotheses must be specified before applying the energy method on an infinite interval.

In a weakly nonlinear expansion of the long-wave convection equation with broken Boussinesq symmetry at $\mu =2$, a critical Fourier mode ${\Theta }_1=A{e}^{ix}+c.c.$ generates ${\Theta }_2=(2is/9)A^2{e}^{2ix}+c.c.$. The quadratic interaction of these first and second harmonics feeds back $8s^2|A{|}^{2}A/9$ into the critical Fourier mode, while the cubic gradient nonlinearity contributes $-3|A{|}^{2}A$. The Fredholm solvability condition is consequently ${\mu }_{2}A-g|A{|}^{2}A=0$. The sign of $g$ distinguishes supercritical and subcritical branches; $g=0$ requires a higher-order amplitude equation.

For a real smooth solution of the long-wave convection equation with broken Boussinesq symmetry, assume a periodic spatial average or an existing long-interval average with vanishing endpoint fluxes. Integration by parts gives the exact energy method identityThus $\mu <{\mu }_E$ prevents growth of the mean-square temperature at arbitrary amplitude in this averaging class. This sufficient nonlinear bound need not equal the linear instability threshold, and it does not imply pointwise monotonicity of the temperature.

Thermal convection is fluid motion driven by temperature-dependent buoyancy. It transports heat by bulk motion in addition to thermal conduction.

A convection roll is a circulating cell pattern whose ideal velocity and temperature fields are periodic across a chosen horizontal direction and independent of the coordinate along the convection roll axis. In a two-dimensional stream function representation, its velocity field is $(-{\psi }_z,0,{\psi }_x)$. Rayleigh-Bénard convection can select stationary or oscillatory convection rolls; a two-dimensional convection roll may be unstable to three-dimensional disturbances, a different convection roll orientation, or slow amplitude and phase modulations. Existence of a saturated Landau amplitude equation branch does not alone establish its stability.

oscillatory convection is time-dependent convection emerging through nonzero-frequency overstability or a Hopf bifurcation. At onset a complex-conjugate growth-rate pair crosses the imaginary axis. Travelling and standing convection rolls are distinct nonlinear possibilities, and their stability cannot be decided from the linear angular frequency alone.

At fixed orientation, opposite travelling amplitudes can obey ${\dot{Z}}_{\pm }=(r+i{\omega }_0)Z_{\pm }-(g|Z_{\pm }{|}^2+h|Z_{\mp }{|}^2)Z_{\pm }$. For $r>0$ and $g_r=\mathrm{Re}g>0$, a travelling branch has one nonzero intensity $r/g_r$ and is stable to its counter-propagating amplitude when $h_r>g_r$. A standing branch has both intensities $r/(g_r+h_r)$ and is stable within the pair when $g_r>h_r$ and $g_r+h_r>0$. These follow by linearizing the real intensity equations. Spatial modulation and other orientations remain separate stability tests.

stationary convection has a time-independent velocity and temperature field, in the chosen fixed frame. Its onset from a conduction state corresponds to a real growth rate passing through zero. Stability of a resulting steady convection roll is a separate nonlinear question.

A stable stratification gives a restoring buoyancy force and positive squared buoyancy frequency for an adiabatically displaced parcel. An adverse specific entropy gradient can instead drive convection; other forces such as rotation can modify the full stability criterion.

Convective overshoot carries material from an unstable convecting region into an adjacent stably stratified region because its momentum does not vanish at the stability boundary. In stars this mixes material beyond the boundary determined by a local stellar convective stability criterion, changing stellar core masses and nuclear lifetimes.

Double-diffusive convection occurs when two properties, such as temperature and composition, affect buoyancy but diffuse at different rates. A stabilizing composition gradient can oppose a destabilizing thermal gradient and change the form of heat transport.

Layered convection consists of mixed convecting layers separated by more weakly transporting interfaces. In a giant planet, composition gradients can support such interfaces and reduce heat loss, contributing to delayed cooling of an inflated giant planet.

Buoyancy is the net force on a body or fluid parcel produced by a pressure gradient in a gravitational or accelerating fluid. Density differences make displaced parcels rise or sink relative to their surroundings.

For a grain of mass density ${\rho }_p$ and volume $\mathcal{V}$ in a fluid of mass density ${\rho }_f$, its effective downward force is $({\rho }_p-{\rho }_f)g\mathcal{V}$, after subtracting buoyancy from weight.

A buoyancy perturbation is the departure $b^{\prime }$ from a prescribed basic buoyancy field. In a fluid with constant buoyancy frequency $N$, vertical velocity $w^{\prime }$ changes it according to $({\partial }_t+\mathbf{U}\cdot \nabla )b^{\prime }=-N^2{w}^{\prime }$.

In a radially stratified shearing sheet, one can normalize the scalar buoyancy perturbation by writing $b_x=-N^2\theta$ and $D_t\theta =u_x+\xi {\nabla }^2\theta$. With this convention $\theta$ has dimensions of length, rather than temperature; $\xi$ is thermal diffusivity.

An isopycnal is a surface of constant mass density. In a Boussinesq approximation, it is equivalently a surface of constant buoyancy.

Hydrodynamic stability studies whether small disturbances to a fluid base state decay, remain neutral, or grow. Linear normal modes reduce this question to eigenvalues or growth rates.

A critical layer is a region where the leading wave phase speed coincides with the local shear-flow velocity and an outer approximation degenerates. Small viscosity or other corrections must then be retained on a distinguished transverse scale. The width depends on the order of vanishing of $U-c_0$ and on the governing mode equation; simple-shear and stationary-profile layers have different balances.

For the Prandtl normal-mode equation with viscosity scaled to one and a profile having a nondegenerate stationary point, $U-c_0\sim (y-y_c{)}^2/2$. A piecewise outer mode proportional to $U-c_0$ has a second-derivative jump. Balancing the advective and third-derivative terms smooths that jump over width ${\alpha }^{-1/4}$, with velocity amplitude ${\alpha }^{-1/2}$ and a phase-speed correction of the same order. Matching selects the eigenvalue through a third-order inner problem.

A stationary point in a non-monotone velocity profile can support a quarter-power critical layer whose phase-speed correction has positive imaginary part. The resulting normal mode grows at a rate proportional to the square root of the wavenumber. Such unbounded high-frequency amplification demonstrates linear instability of the frozen layer; nonlinear evolution and precise well-posedness consequences require separate analysis.

For a wave with zonal phase velocity $c$ in a basic shear $U(z)$, a critical level is a height $z_c$ at which $U(z_c)=c$. The Doppler-shifted material frequency then vanishes. This means equality of the laboratory velocities; the $U$ and $-c$ terms cancel in the wave frame. It does not mean $U=-c$. A semi-infinite Eady edge wave with zero interior potential vorticity anomaly can remain regular through its steering height.

Baroclinic instability converts potential energy of a stratified sheared basic flow into growing disturbances. In the Eady model, coupled Boundary Rossby waves provide a useful mechanism: their induced velocities displace boundary buoyancy anomalies, and suitable wavelengths permit phase locking. A single neutral boundary wave does not by itself imply exponential instability.

A high-Reynolds number long-wave mode near a wall has a viscous layer of thickness $\delta =(\alpha R{)}^{-1/3}$, an order-one region following the base profile, and a pressure-decay region of thickness $1/\alpha$. The lower Airy function solution fixes velocity displacement and fluid pressure. The middle Rayleigh equation for inviscid shear flow transports these to the outer constant-velocity region. Matching its decaying pressure produces an Airy wall-layer matching determinant. For order-one amplitudes, constant leading middle pressure requires ${\alpha }^2\ll \delta$; the distinguished balance $\alpha \sim \delta$ gives $\alpha \sim R^{-1/4}$.

With the Airy displacement integrals, the lower pressure-to-velocity ratio is $-iq\mathrm{Ai}\prime (-qC)/I_0(C)$. The decaying upper mode gives that ratio as $\alpha /\delta$. Hence for $\alpha =k{R}^{-1/4}$ the dispersion relation is $k^{4/3}{I}_0(C)+iq\mathrm{Ai}\prime (-qC)=0$. Writing it without division remains meaningful if an individual moment vanishes. The wavespeed is $c=\delta C$ and a positive wavenumber grows temporally when $\mathrm{Im}C>0$.

For an incompressible parallel flow with base velocity $U(y)$ and perturbation proportional to $e^{i\alpha (x-ct)}$, divergence of the linearized momentum equations gives $p\prime \prime -{\alpha }^{2}p=-2i\alpha U\prime v$. At a no-slip boundary condition, streamwise momentum gives $p(0)=u\prime \prime (0)/(i\alpha R)$. The equation encodes how base shear generates perturbation fluid pressure.

Relative to the kinetic-energy inner product, integration by parts reverses incompressible base transport, transposes the velocity-gradient coupling, and preserves viscous diffusion. With reverse time $\tau =-t$, the adjoint base state is evaluated at $-\tau$. The Leray-Helmholtz projection or adjoint pressure enforces divergence-free adjoint velocity. The curl of a cross product and vorticity cross-product identity give the equivalent form ${\partial }_{\tau }v=\Omega \times v-\nabla \times (U\times v)-\nabla \pi +{\mathrm{Re}}^{-1}\Delta v$.

For a constant reference density, fluid perturbation kinetic energy is proportional to $\Vert u{\Vert }^2/2$ under the displayed $L^2$ inner product. An adjoint operator for perturbation evolution must be defined relative to this inner product, including boundary conditions and the projection onto incompressible flow.

For the linear complex Ginzburg-Landau equation with real $U,c_d,\mu$ and positive real diffusion coefficient scaled to one, the maximum temporal growth rate is $\mu$ and the absolute growth rate is $\mu -U^2/[4(1+c_d^2)]$. For $U>0$, $\mu <0$ is stable, $0<\mu <{\mu }_a$ has convective hydrodynamic instability, and $\mu >{\mu }_a$ has absolute hydrodynamic instability. Both equality boundaries are marginal in exponential rate.

On a ray $x/t=V$, a localized wave packet is governed by a physically selected saddle point satisfying $d\omega /dk=V$ in its analytic dispersion relation. The exponent's real growth is $\mathrm{Im}({\omega }_{\star }-V{k}_{\star })$. The fixed-frame case $V=0$ gives the absolute growth rate. This ray formulation makes the frame dependence of convective hydrodynamic instability explicit.

A convective hydrodynamic instability amplifies an advected localized disturbance while its amplitude decays at any fixed position. The distinction from absolute hydrodynamic instability depends on the observation frame.

An absolute hydrodynamic instability makes the response to a localized impulse grow at a fixed spatial point in a specified frame. This differs from convective hydrodynamic instability, which amplifies a moving wave packet while its response decays at fixed positions. For analytic dispersion relations, the relevant absolute frequency is associated with an accessible zero-group velocity saddle, with the spatial-branch selection checked rather than assumed for an arbitrary dispersion relation.

The absolute frequency is the generally complex angular frequency at the physically selected absolute wavenumber. Its imaginary part is the fixed-position exponential growth rate under the convention $e^{-i\omega t}$. Marginal zero exponential growth may still carry an algebraic prefactor.

A time-independent ray Hamiltonian conserves the stationary-observer absolute frequency. A horizontally uniform background also conserves horizontal wavenumber. The intrinsic frequency $\hat{\omega }=\omega -kU(z)$ nevertheless changes when the ray moves through shear. These follow from the Hamiltonian ray-tracing equations and are essential to identifying a critical level of an internal gravity wave.

The absolute growth rate is the exponential growth rate of a localized impulse at a fixed position in the chosen observation frame. Under the convention $e^{i(kx-\omega t)}$, it is the imaginary part of the physically selected absolute frequency. Positive values give absolute hydrodynamic instability; negative values can still coexist with convective hydrodynamic instability.

An absolute wavenumber is a complex wavenumber at the selected zero-group velocity saddle of a dispersion relation. For the linear complex Ginzburg-Landau equation, $k_0=-U/[2(c_d-i)]$; the positive real part of the diffusion coefficient makes the Gaussian saddle accessible from the real Fourier contour.

The Orr-Sommerfeld equation governs the wall-normal velocity of normal mode disturbances to an incompressible viscous parallel shear flow. Here $D=d/dy$, ${\kappa }^2={\alpha }^2+{\beta }^2$, and $Re$ is the Reynolds number. It follows by eliminating pressure from the linearized Navier-Stokes equations. At a rigid no-slip wall, $\hat{v}=D\hat{v}=0$. In the inviscid limit it reduces to the Rayleigh equation for inviscid shear flow.

For nondimensional constant shear $U=y$ and streamwise wavenumber one, put $W=(D^2-1)\hat{v}$. The Orr-Sommerfeld equation gives $W^{\prime \prime }=[1+iRe(y-c)]W$. The displayed substitution transforms it into the Airy ordinary differential equation, because $(e^{i\pi /6}{)}^3=i$. The scale $R{e}^{-1/3}$ balances shear advection against viscous diffusion.

The equation $(D^2-1)\hat{v}=W$ is inverted by the displayed formula: differentiating twice produces the endpoint contribution $W(y)$, since $\sinh 0=0$ and $(\sinh {)}^{\prime }(0)=1$. If $W=a_1\mathrm{Ai}(Z)+a_2\mathrm{Bi}(Z)$, there are four independent constants $C_{+},C_{-},a_1,a_2$. This reconstructs the velocity from the Airy reduction of the Orr-Sommerfeld equation in constant shear; four no-slip and no-flux conditions determine the boundary eigenproblem.

The Squire equation governs wall-normal vorticity $\eta ={\partial }_{z}u-{\partial }_{x}w$ in the same normal mode convention as the Orr-Sommerfeld equation. It follows by taking ${\partial }_z$ of the streamwise momentum equation minus ${\partial }_x$ of the spanwise momentum equation. At a rigid no-slip wall, $\hat{\eta }=0$.

A Squire mode has zero wall-normal velocity and nonzero wall-normal vorticity, satisfying the homogeneous Squire equation. For real base velocity, positive finite Reynolds number, and zero boundary terms,for a nontrivial mode in a finite channel with no-slip walls. This follows by multiplying the homogeneous equation by $\overline{\eta }$, integrating by parts and taking the real part. Such modes are damped even when the coupled velocity-vorticity system can exhibit transient growth from non-normal modes.

An Orr-Sommerfeld mode has nonzero wall-normal velocity satisfying the Orr-Sommerfeld equation. Its wall-normal vorticity satisfies the accompanying forced Squire equation. The triangular coupling separates the velocity eigenproblem from the vorticity forcing; the vorticity need not vanish in three dimensions.

Inertial instability occurs when displaced fluid parcels amplify their displacement by conserving absolute momentum in a rotating flow. For a uniformly stratified basic state on an f-plane, a necessary condition is that the Ertel potential vorticity have the opposite sign to the Coriolis parameter.

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

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.

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.

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 $J$.