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Exchange two equal fluid elements of volume $V$ without thermal diffusion in a Boussinesq approximation. Each retains its buoyancy. If the separation is $(\delta y,\delta z)$ and the background buoyancy difference is $\Delta b=B\delta y+N^2\delta z$, the pair's gravitational potential energy changes by ${\rho }_{0}V\delta z\Delta b$. For $B<0$, $N^2>0$ and poleward-rising slope $s=\delta z/\delta y$, this is negative exactly when $0<s<-B/N^2$. The minimum at fixed $\delta y$ is attained at half the isopycnal slope. This energetic test alone is not sufficient for dynamical baroclinic instability: rotation and constraints on parcel motion must also be considered.

The Eulerian time derivative measures a continuum field's rate of change at a fixed Eulerian coordinate. For a velocity field $\mathbf{u}$, the material derivative instead follows a fluid element and satisfies $Df/Dt={\partial }_{t}f+\mathbf{u}\cdot \nabla f$. A nonuniform steady flow has zero Eulerian time derivative of velocity but can have nonzero material acceleration.

A fluid element is an infinitesimal material parcel followed along a Lagrangian trajectory.

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.

Fluid mechanics studies the motion and forces of liquids and gases. Its central simplification is the continuum approximation: replace individual molecules by bulk properties defined at every position and time. The averaging region must contain many molecules while remaining small compared with the distances over which those properties vary. A fluid element is such a small material parcel, rather than a single molecule. This connects fluid mechanics to continuum mechanics and makes calculus useful for describing flow. ^[ref]

Three basic fields are the velocity field $\mathbf{u}(\mathbf{x},t)$, the mass density $\rho (\mathbf{x},t)$ and the pressure $p(\mathbf{x},t)$. The first gives a parcel's instantaneous velocity; the second gives mass per unit volume. The pressure contributes an inward normal traction $-p\mathbf{n}$ across a surface with outward unit normal $\mathbf{n}$. Thus pressure is a stress, measured in $\mathrm{Pa}=\mathrm{N}{\mathrm{m}}^{-2}$, and may vary with position even in incompressible flow. Additional viscous stresses can act tangentially. ^[ref]

The guiding questions are what must be conserved and which forces matter. Mass conservation constrains the velocity field; momentum conservation determines its response to pressure gradients, viscosity and body forces. An incompressible flow model, a Reynolds number estimate and Bernoulli's principle answer different parts of a problem. Their appeal is that a few carefully stated assumptions can determine useful quantities before the entire velocity field is known.

A gravity-capillary wave has restoring contributions from both gravity and surface tension. Consider an initially resting, horizontally unbounded liquid layer of constant depth $h$, with uniform mass density $\rho$, incompressible flow, inviscid flow and irrotational flow. The upper gas exerts constant pressure and has negligible inertia. Assume a clean interface with constant surface tension $\gamma$: no surfactant stresses, imposed current or wind. The wave amplitude $a$ must satisfy $ka\ll 1$ and $a\ll h$, so linearized boundary conditions apply. Deep water means $kh\gg 1$, rather than a particular depth in metres. ^[ref]

Let $z=0$ be the undisturbed free surface, with liquid below it. Potential flow gives a velocity potential $\varphi$ with $\mathbf{u}=\nabla \varphi$, and incompressible flow gives the Laplace equation. The linearized kinematic boundary condition for a free-surface graph and the Unsteady Bernoulli equation, including gravity and the Young–Laplace equation, giveHere $\eta (x,t)$ is the interface displacement; the signed small-slope pressure jump is $p_{\mathrm{liquid}}-p_{\mathrm{gas}}=-\gamma {\eta }_{xx}$. A wave crest has ${\eta }_{xx}<0$, so its capillary pressure is positive. Choose a right-travelling Fourier mode of positive wavenumber $k$ and angular frequency $\omega$:The factor $e^{kz}$ decays into the deep liquid, satisfies the Laplace equation, and makes ${\varphi }_z={\eta }_t$ at $z=0$. Substituting in the dynamic boundary condition leavesThis derivation displays the two restoring contributions with the same sign. Dimensional analysis also checks the result: $\gamma /\rho$ has units ${\mathrm{m}}^3{\mathrm{s}}^{-2}$, so both $gk$ and $(\gamma /\rho )k^3$ have units ${\mathrm{s}}^{-2}$. ^[ref]

At finite depth, a flat impermeable bottom at $z=-h$ requires ${\varphi }_z=0$. Replacing $e^{kz}$ by the normalized vertical factor $\cosh (k(z+h))/\cosh (kh)$ makes the surface ratio ${\varphi }_z/\varphi =k\tanh (kh)$. Eliminating the mode amplitudes then givesThus the same model supplies its own depth check: use the deep-water formula only when $\tanh (kh)$ is sufficiently close to one for the required accuracy. For example, $kh=3$ reduces $\omega$ by about $0.248\%$ relative to deep water, while $kh=1$ reduces it by about $12.7\%$. These percentages are calculations from this factor. ^[ref]

In deep water the phase velocity magnitude follows a wave crest, while the group velocity magnitude follows the wave envelope of a narrow wave packet. Differentiating the positive-frequency dispersion relation givesFor $k{\ell }_c\ll 1$, the surface gravity wave limit has $c_g\simeq c_p/2$. For $k{\ell }_c\gg 1$, the deep-water capillary-wave dispersion relation has $c_g\simeq 3c_p/2$: a wave crest slips backward relative to the faster wave envelope, even while both advance. Neither speed is the oscillating fluid element's own velocity. ^[ref]

There is a particularly neat coincidence at the crossover. Differentiate $c_p^2=g/k+\gamma k/\rho$: its unique minimum is at $k_c=\sqrt{\rho g/\gamma }=1/{\ell }_c$, exactly where the restoring terms are equal. ThereforeFor the stipulated parameters in capillary length, the minimum is $0.231\mathrm{m}{\mathrm{s}}^{-1}$. At $\lambda =3\mathrm{mm}$, direct substitution gives $f=\omega /(2\pi )\simeq 131\mathrm{Hz}$, $c_p\simeq 0.394\mathrm{m}{\mathrm{s}}^{-1}$ and $c_g\simeq 0.580\mathrm{m}{\mathrm{s}}^{-1}$. This short wavelength is capillary dominated, but its group velocity is still slightly below the pure-capillary ratio $3c_p/2$ because gravity has not vanished.

An incompressible flow preserves the volume of each moving fluid element. Its local condition is zero velocity divergence, $\nabla \cdot \mathbf{u}=0$. Combined with mass conservation, it says that each parcel's mass density stays unchanged as it moves. This does not require every parcel to have the same mass density: an initially nonuniform distribution can be transported without compression. Nor does incompressibility require a constant velocity or constant pressure.

For positive mass density, the continuity equation makes the equivalence precise:The material derivative follows a fluid element. Thus $D\rho /Dt=0$ if and only if the velocity field is a divergence-free vector field. The distinction between following a parcel and observing a fixed position prevents confusing incompressibility with an unchanging density field. ^[ref]

For a rigid pipe with impermeable walls, no leaks and steady flow, mass conservation at constant mass density implies a constant volumetric flow rate:Here $\mathbf{n}$ points downstream, $A$ is the section's area and $\overline{U}$ is the mean normal velocity, not necessarily the centreline speed. No assumption of inviscid flow or a uniform velocity profile is needed. If unequal amounts entered and left a fixed segment, its mass would accumulate, contradicting steady flow.

For a worked example, take horizontal water flow with stipulated mass density $1000\mathrm{kg}{\mathrm{m}}^{-3}$, inlet radius $R_1=0.010\mathrm{m}$, outlet radius $R_2=0.005\mathrm{m}$ and inlet mean velocity ${\overline{U}}_1=0.20\mathrm{m}{\mathrm{s}}^{-1}$. The circular areas giveThe radius halves, the area falls to one quarter, and the mean velocity quadruples. The mass flow rate is $\rho Q=0.02\pi \mathrm{kg}{\mathrm{s}}^{-1}$. The same amount of water passes both sections each second; narrowing changes its speed rather than compressing it. The Reynolds number and Bernoulli's principle discussions use this same example to examine viscous effects and the additional assumptions needed to calculate pressure.

The Lagrangian pressure perturbation is the change in pressure experienced by a displaced fluid element: $\Delta p=\delta p+\mathbf{\xi }\cdot \nabla p_0$. For a free surface against fixed external pressure, $\Delta p=0$, rather than $\delta p=0$ at the original surface.

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.

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.

The first argument mixes a material fluid element with a fixed spatial volume. For a material volume element following the fluid, its volume is generally time dependent:Therefore the material derivative cannot be taken outside the product with $\delta V$ as in the proposed cancellation. Instead, mass conservation statesThe momentum balance for a material fluid element, using Newton's second law, is consequentlyFor a fixed control volume, fluid crosses its boundary and carries momentum; a momentum flux term must be included. Its local conservative form is ${\partial }_t(\rho \mathbf{u})+\nabla \cdot (\rho \mathbf{u}\otimes \mathbf{u})=-\nabla p$, equivalent to the displayed Euler equations after using the continuity equation. The erroneous equation omits the term $\rho \mathbf{u}\nabla \cdot \mathbf{u}$ needed to convert a material derivative of momentum density into this conservative form.

For the second task, take a homogeneous fluid with constant mass density $\rho$, prescribed gravitational acceleration $\mathbf{g}$, incompressible flow and inviscid flow. Its equations areUnder the replacement $\rho \mapsto \tilde{\rho }=C\rho$, the same velocity solves these equations if $p\mapsto \tilde{p}=Cp$, up to a spatially constant pressure offset. This is density scaling invariance of incompressible Euler flow:The comparison assumes the same geometry and compatible velocity or gravitational free surface conditions; an independently specified nonuniform boundary pressure would supply additional information. The gravitational field is taken as prescribed. If the fluid's own self-gravity must be included, its field changes with mass density, so the invariance need not hold. A single freely moving particle's density-independent gravitational acceleration does not justify neglecting that distinction.

Let $V(t)$ be any material volume, moving with the velocity field $\mathbf{u}$. The Reynolds transport theorem in the required form isapplied componentwise when $f$ is vector-valued. Mass conservation givesSince the material volume is arbitrary, the local continuity equation follows:For an inviscid fluid with no body force, the balance of linear momentum equates the rate of change of total momentum to the boundary pressure force:using the divergence theorem. The integrand on the left expands asThe bracket vanishes by mass conservation, leaving the Euler momentum equationUnlike the incompressible flow case, the mass density need not be constant along a fluid element.

Write $\rho$ for the mass density. A material volume contains the same fluid elements at all times, so conservation of mass requiresby the Reynolds transport theorem. Since the initial material volume can be chosen arbitrarily, continuity of the integrand forces it to vanish pointwise. Thus the continuity equation isFor positive mass density, the incompressibility condition $D\rho /Dt=0$ reduces this to $\nabla \cdot \mathbf{u}=0$. Mass conservation itself does not require incompressible flow.

For a scalar fluid property $f(\mathbf{x},t)$, the Eulerian time derivative ${\partial }_{t}f$ measures change at a fixed Eulerian coordinate. The material derivative, also called the Lagrangian time derivative, measures change while following one fluid element. If its Lagrangian trajectory is $\mathbf{X}(\mathbf{a},t)$, where $\mathbf{a}$ is its fixed Lagrangian coordinate, then ${\partial }_t\mathbf{X}=\mathbf{u}(\mathbf{X},t)$.

The chain rule therefore givesThe last term accounts for entering a region where the property differs. For example, a spatially varying but time-independent velocity field has zero Eulerian time derivative, while a moving fluid element can have nonzero material acceleration. The same formula applies componentwise to a vector field.

Consider the nonresonant traveling forced solution, with $k>0$ and $c\ne 0$. Put $x=\bar{x}+\xi$, $y=\bar{y}+\upsilon$, where $(\bar{x},\bar{y})$ is the mean position of a fluid element. At linear order, evaluate the velocity field at this mean position: substituting $\xi ,\upsilon$ into its spatial arguments would change the already first-order velocity only at second order. With $\Theta =k(\bar{x}-ct)$, differentiation of the velocity potential givesSince $\dot{\Theta }=-kc$, integration, with zero mean displacements, yieldsIf both coefficients are nonzero, eliminating the wave phase proves that the pathline is an ellipse:Its principal axes of an ellipse are horizontal and vertical, with respective semi-axesThus the full axis lengths are $2|A|$ and $2|B|$. The signed coefficients also determine the direction of traversal. The useful checks $B(0)=b$ and $B(-d)=a$ reproduce the vertical displacements of the free surface and bottom.

To identify the major axis without comparing the two expressions separately, use ${\cosh }^{2}s-{\sinh }^{2}s=1$:Consequently the major axis is horizontal if $c^2<g/k$ and vertical if $c^2>g/k$; equality gives circles. If $A$ or $B$ vanishes at a particular depth, the first-order orbit degenerates to a line segment. These are linearized particle orbits of a bottom-forced gravity wave, not exact closed trajectories at all orders: a higher-order calculation may introduce a mean drift.

The velocity of a fluid element is the time derivative of its position with its Lagrangian coordinates $X,Y$ held fixed. Direct differentiation givesReplacing the Lagrangian coordinates by the current Eulerian coordinates therefore gives the velocity fieldThe distinction between following a fluid element and differentiating at fixed Eulerian coordinates is what connects the given deformation map to this velocity field.

Compressing a small region raises its mass density and, for positive compressibility, its pressure. The outward pressure gradient force accelerates neighbouring fluid elements. Their outward motion compresses fluid farther away, carrying the disturbance outward. The original compressed region relaxes; inertia allows the fluid motion to continue beyond zero excess pressure, producing a rarefaction and a restoring pressure gradient force. Elastic compression and inertia therefore support a propagating longitudinal wave.

In a local planar description with acoustic displacement $\xi (x,t)$,so ${\xi }_{tt}=c_0^2{\xi }_{xx}$. The speed of sound is a propagation speed, whereas the acoustic particle velocity is the small back-and-forth motion of the medium. In linear acoustics, energy travels with the disturbance without first-order net transport of the fluid over a wave period.

Taking the imaginary part of the complex potential gives the streamfunctionThe corresponding velocity field isPut $\zeta =z{e}^{i\alpha /2}=\xi +i\eta$. Then $w=\beta {\zeta }^2$ and $\psi =2\beta \xi \eta$. Thus the streamlines are the previous rectangular hyperbolas rotated clockwise by $\alpha /2$. Their separatrices have angles $\theta =-\alpha /2+n\pi /2$; the extension axis is $\theta =-\alpha /2$ and the compression axis is perpendicular to it. The middle panel of the figure shows this rotation.

For pure advection, a marked fluid element obeysConsequently the dye circle becomes an ellipse with semiaxes $R{e}^{2\beta t}$ and $R{e}^{-2\beta t}$. Its area isStretching in one principal direction exactly compensates compression in the other. Equivalently the flow-map Jacobian determinant is one, as required by incompressibility. Negligible diffusion is needed to regard the sharp dye boundary as a material curve.

Choose $y$ northward and $z$ upward in the Northern Hemisphere. In a locally linear Boussinesq approximation, write the background buoyancy as $\bar{b}=b_{\ast }+By+N^{2}z$, with horizontal buoyancy gradient $B<0$ and positive squared buoyancy frequency $N^2$. The stable density stratification resists a purely vertical displacement, but the isopycnals slope upward poleward:The geopotential surfaces are horizontal in this local approximation. Their inclination differs from that of the isopycnals, which is the relevant baroclinicity.

Exchange two equal small fluid elements, each of volume $V$, between positions separated by $(\delta y,\delta z)$. The adiabatic parcel exchange has no thermal diffusion, so each parcel retains its buoyancy. The background buoyancy difference is $\Delta \bar{b}=B\delta y+N^2\delta z$. Because the variable-density contribution to gravitational potential energy is $-{\rho }_{0}Vbz$, the change in the pair's gravitational potential energy isFor a poleward-rising exchange, set $\delta z=s\delta y$ with $s>0$. ThenThe trajectory lies between a geopotential surface and an isopycnal. The poleward-rising parcel starts more buoyant than the parcel it replaces, and the equatorward-sinking parcel starts less buoyant: the buoyancy does positive work on both. This releases available potential energy into kinetic energy, allowing eddies to amplify if the full rotating dynamics supports the displacement and dissipation does not remove the gain.

A trajectory steeper than the isopycnals, $s>s_{\rho }$, instead costs gravitational potential energy; an equatorward-rising trajectory also costs it. Motion exactly along a geopotential surface or an isopycnal gives zero change in this ideal exchange. At fixed $\delta y$, the largest release occurs at $s=s_{\rho }/2$, since $N^{2}s(s-s_{\rho })$ is minimized there.

This is an energetic possibility argument for baroclinic instability, not a sufficient normal mode instability criterion. In particular, unrestricted parcel motion must not be assumed in a rotating fluid: conservation of angular momentum can impose an additional restoring effect. Symmetric instability has a separate Ertel potential vorticity criterion; ordinary Eady instability can occur without violating that criterion.

The quasi-geostrophic potential-vorticity equation is the leading balanced form of potential-vorticity conservation for an inviscid rotating fluid with stable density stratification, small Rossby number and an adiabatic process for each fluid element. Its parent conservation law is Ertel potential vorticity conservation. Horizontal geostrophic balance, vertical hydrostatic balance, and the beta-plane approximation reduce it to advection of one scalar by the leading horizontal geostrophic flow.

One can see its origin by combining the vertical vorticity equation with the thermal equation. Set $D_g={\partial }_t+{\mathbf{u}}_g\cdot {\nabla }_h$, ${\mathbf{u}}_g=(-{\psi }_y,{\psi }_x)$, and ${\zeta }_g={\nabla }_h^2\psi$. The quasi-geostrophic approximation givesHere $b$ is the buoyancy perturbation relative to the reference stable density stratification, $N=N(z)$ its buoyancy frequency, and $w$ the small vertical velocity. The stretching of absolute vorticity in the first equation is offset by the change of stratification represented by the second. More precisely,The last dot product is zero because thermal-wind balance makes ${\partial }_z{\mathbf{u}}_g$ perpendicular to ${\nabla }_{h}b$. Adding therefore yieldsThe constant $f_0$ has no effect on the tendency, but its inclusion makes the connection with absolute vorticity clear. Vertical advection of $q$ is beyond the retained order; the vertical velocity has already supplied the stretching and thermodynamic terms.

For a simple balanced weather forecast, the essential cycle is diagnose, advect, invert:

This is an elliptic boundary value problem, not a local algebraic diagnosis. Its solution supplies the next geostrophic flow and, through $b=f_0{\psi }_z$, the next thermal field. Repeat the cycle to move the balanced weather systems forward.

Interior potential vorticity alone is insufficient: boundary thermal anomalies can induce flow even with zero interior anomaly, as in the Eady model. The method filters fast inertia-gravity waves and is useful for large-scale extratropical weather, while tropical dynamics, convection, moisture, friction and diabatic heating require additional physics. If vertical motion is wanted, the quasi-geostrophic omega equation supplies a further diagnostic calculation.