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Assume a steady, small-Rossby number surface boundary layer, neglect horizontal viscosity and nonlinear acceleration, take the pressure gradient to be independent of depth, impose no normal flow at the surface, and let the viscous stress vanish at $z=-h$. Subtract the depth-independent geostrophic balance from horizontal momentum. For the ageostrophic velocity,Integrating from $-h$ to $0$ and usinggives the Ekman transport

Depth-integrated mass conservation, with $w(0)=0$, givesHence the vertical velocity entering the ocean interior is the Ekman pumping velocityFor constant $f$ this reduces to

Put $D(x,y)=H(x,y)-h(x,y)$ and let $\mathbf{u}=(u,v)$ be the depth-independent interior velocity. The kinematic boundary conditions on the sloping upper and lower surfaces areIntegrating mass conservation through the layer givesThe inviscid vertical-vorticity equation isConsequently the shallow-water potential vorticityobeys the forced evolution equationPositive upward Ekman pumping removes layer thickness and raises the potential vorticity of the remaining column.

For a steady small-Rossby number flow, $q\simeq f/D$, soEquivalently,The same result follows from the integrated vortex-stretching balance

When $\nabla D=(c,0)$ with $c>0$,If the upper pumping is absent or weak and the upper surface is locally level, the impermeable-bottom condition is $w_b=-cu$. The stipulated $w_b<0$ then gives $u>0$, and in the Northern HemisphereThe steady interior flow is therefore directed northeastward, along contours of $f/D$ in the unforced limit. As a parcel moves eastward into deeper water, its vortex column stretches; a poleward displacement increases $f$ and preserves potential vorticity. Nonzero Ekman pumping drives motion across the $f/D$ contours. This is topographic potential-vorticity steering and the associated topographic Sverdrup balance.

Define the residual-circulation stream function byThis satisfies residual mass conservation identically. LetThe momentum equation is

Differentiate geostrophic balance vertically and hydrostatic pressure meridionally to obtain thermal-wind balanceAfter a time derivative, the transformed density equation givesOn the other hand, the $z$ derivative of momentum givesEliminating ${\overline{u}}_{tz}$ yields the Eliassen equation for residual circulation

Local mass conservation of each particle species gives the system of conservation lawsFor $\mathbf{\varphi }=({\varphi }_1,{\varphi }_2{)}^T$ and flux $F_i={\varphi }_i{W}_i$, the flux Jacobian isWriting its entries as $A_{ij}$, the two characteristic speeds are its eigenvaluesThe discriminant is nonnegative because $A_{12}{A}_{21}={\varphi }_1{\varphi }_2{\hat{W}}_1{\hat{W}}_2\ge 0$, so positive concentrations of two settling species give real characteristics.

Write the density as a hydrostatic reference profile $\hat{\rho }(z)$ plus a small perturbation ${\rho }^{\prime }$, and define buoyancy and buoyancy frequency byThe Boussinesq approximation to the Navier-Stokes equation, together with mass conservation, iswhere $D/Dt$ is the material derivative, $\nu$ is kinematic viscosity, and $\kappa$ is mass diffusivity. Linearizing about rest and eliminating pressure and buoyancy givesFor a plane wave proportional to $e^{i(kx+mz-\omega t)}$, with $K^2=k^2+m^2$, the viscous-diffusive dispersion relation isIn the inviscid limit this becomesFor weak diffusion the two oscillatory roots areso a freely evolving Fourier mode decays.

A single plane wave is also an exact solution of the nonlinear equations. Every field depends only on its phase $\Theta =\mathbf{k}\cdot \mathbf{x}-\omega t$, while incompressible flow gives $\mathbf{u}\cdot \mathbf{k}=0$. Therefore $\mathbf{u}\cdot \nabla$ annihilates both $\mathbf{u}(\Theta )$ and $b(\Theta )$, and all nonlinear advection terms vanish.

For a boundary-forced wave with real $\omega$, nonzero $\nu$ or $\kappa$ instead makes the bulk vertical wavenumber complex, attenuating the propagating beam. Because diffusion raises the spatial order of the equations, additional short vertical-wavenumber roots form viscous and scalar boundary layers; they allow a no-slip velocity condition and a scalar no-flux condition to accompany impermeability. These layers and bulk attenuation become essential near critical internal-wave reflection, where the inviscid reflected wavelength collapses.

The Boussinesq approximation replaces density by a common reference value ${\rho }_0$ in inertia, mass conservation, and pressure acceleration, while retaining small density differences in the gravitational buoyancy term. It gives incompressible flow and is appropriate here wheneven though those small differences drive the room-scale motion. It would fail for order-one thermal density contrasts or strongly compressible ventilation.

The Batchelor entrainment hypothesis sets the mean inflow speed across a turbulent plume edge to $\alpha$ times a representative axial plume speed, where $\alpha$ is the entrainment coefficient. It closes integral plume balances by relating plume growth to its speed. Applied here, it produces an entraining axisymmetric warm plume above the floor source and a one-sided cold wall line plume below the vent. Treating both as turbulent top-hat plume models neglects source regions, detailed profiles, wall friction, interaction between the two plumes, and the finite thickness of the density interface; these are the principal modelling assumptions.

Steady mass conservation with inward-positive accretion rate givesWrite the specific angular momentum as $l=R^2\Omega$. Multiplying the azimuthal equation by $2\pi R$ and using the mass equation shows that the sum of advected and viscous angular-momentum flux is constant:The right-hand side implements the zero-torque inner boundary condition. For a Keplerian accretion disk, $l\propto R^{1/2}$ and $R^{3}d\Omega /dR=-(3/2)l$. ThereforeandFar outside the inner edge, $\Sigma \simeq \dot{m}/(3\pi \nu )$ and $u_R\simeq -3\nu /(2R)$.

Steady mass conservation gives $\dot{m}=-2\pi R\Sigma u_R$. With specific angular momentum $l=R^2\Omega$, multiply the angular-momentum equation by $2\pi$ and define the signed viscous torque in an accretion diskThenIntegration from $R_1$ to $R_2$ gives

The viscous power generated in an annulus is $2\pi R{F}_{\mathrm{diss}}dR=\mathcal{G}d\Omega$. Taking the inner torque to vanish and writing $l_{\mathrm{in}}=l(R_{\mathrm{in}})$ gives $\mathcal{G}=-\dot{m}(l-l_{\mathrm{in}})$, henceIntegration by parts yieldsFor steady circular force balance, $de=\Omega dl$; equivalently, the stated equality of gravitational- and rotational-potential differences makes the integral $e_{\mathrm{out}}-e_{\mathrm{in}}$. Therefore

When $R_{\mathrm{out}}\gg R_{\mathrm{in}}$, the outer energy and boundary term vanish. For an approximately Keplerian inner orbit, $e_{\mathrm{in}}=-R_{\mathrm{in}}^2{\Omega }_{\mathrm{in}}^2/2$, soComparison with part (c) givesA Keplerian inner flow generates the standard thin-disk power. A pressure-supported sub-Keplerian slim disk generates less through shear, and its emergent luminosity can be smaller still because photon trapping in an accretion flow carries part of that generated energy through the inner edge.

The galaxy mass--metallicity relation is the observed tendency for more massive galaxies to have larger gas-phase metallicity and stellar metallicity. Gas metallicity is commonly inferred from nebular emission-line ratios in star-forming H II regions, often quoted as $12+{\log }_{10}(\mathrm{O}/\mathrm{H})$ and measured within a finite spectroscopic aperture. Stellar metallicity comes from stellar absorption features or population-synthesis fits and is luminosity weighted unless the analysis explicitly reconstructs a mass-weighted distribution. Galaxy stellar mass is inferred from photometry or a spectral-energy-distribution fit and depends on the adopted initial mass function. Radial metallicity gradients, dust, line calibration, and aperture selection must consequently be matched before samples are compared.

The usual physical explanation is that a shallow potential well lets a low-mass galaxy lose a larger fraction of newly synthesized metals in galactic outflows. The closed-box model of galactic chemical evolution is therefore replaced by a leaky-box model of galactic chemical evolution withwhere $\lambda$ is the mass-loading factor. Under the instantaneous recycling approximation, let $y_z$ be the stellar yield, absorb the returned mass fraction into the definitions, and suppose the escaping gas has the current gas metallicity $Z_g$. Then mass conservation and metal conservation areSubstitution of the first equation into the second cancels the terms that merely transfer pre-existing metals and leavesBecause the mass of metals locked into stars obeys $d{M}_{z,\ast }=Z_{g}d{M}_{\ast }$, integration gives $M_{z,\ast }=M_{\ast }{Z}_{\ast }=\int Z_{g}d{M}_{\ast }$. The total newly made metal mass is partitioned between present gas, stars, and the outflow:Writing the gas-to-stellar mass ratio as $r_g=M_g/M_{\ast }$ therefore producesThus simultaneous gas and stellar metallicities, together with the gas fraction and an assumed nucleosynthetic yield, estimate the integrated mass loading. The corresponding effective yield is $y_{\mathrm{eff}}=y_z/(1+\lambda )$, and the leaky box has

The G-dwarf problem is that the local Milky Way disk contains far fewer low-metallicity long-lived G dwarfs than the constant-yield closed-box metallicity distribution predicts. A yield that rises with metallicity may initially sound promising because enrichment would accelerate after the first generations. In fact it worsens the problem. In a closed box, $d{M}_{\ast }=-d{M}_g$ andFor $y_z=kZ$ and a nonzero initial metallicity $Z_0$ at gas mass $M_{g,0}$,Hence the cumulative mass of stars born below metallicity $Z$ isIf the system begins at $Z_0=0$, the assumed yield also vanishes and enrichment never starts. For $Z_0>0$, the metallicity distribution function haswhich puts still more stellar mass near the low-metallicity floor. Metal-poor gas inflow, pre-enrichment, and selective outflow are therefore more plausible ingredients in resolving the G-dwarf problem.

In a steady axisymmetric outflow, mass conservation makes $2\pi r\Sigma (r)v_r(r)$ independent of $r$. The steady radial dust outflow therefore hasIt has an integrable $(r-a_p{)}^{-1/2}$ pile-up just outside the source ring, where the radial speed starts from zero, and approaches $Sigma\propto r^{-1}$ at large radius.

For a homologous star, hydrostatic equilibrium, mass conservation, and the ideal gas equation give the central scalingswhere $\mu$ is the mean molecular weight. Integrating the nuclear energy-generation law over a fixed homologous profile givesRadiative stellar structure gives independentlyEquating the two luminosities yieldsAt fixed zero-age composition, the stars are therefore homologous with

The effective temperature satisfies $L=4\pi R^2\sigma T_e^4$, so $T_e\propto M^{3/8}$. The zero-age main sequence consequently hasIt is a steep line rising toward high luminosity and high temperature on a Hertzsprung-Russell diagram.

For fully ionized hydrogen and helium with $Y=1-X$,while $\kappa \propto 1+X$. At fixed mass,and the corresponding radius relation is

At the pure-hydrogen zero-age point $X=1$,whereasHenceandAs hydrogen is consumed, $X$ falls, so both $L$ and $T_e$ rise. In the usual diagram with temperature increasing leftward, the evolutionary track initially moves upward and leftward from the zero-age main sequence.

Let $z$ increase downward. The slender viscous thread is locally in uniaxial extension. The transverse stress equals the ambient pressure, so the Trouton ratio gives the excess axial stress ${\sigma }_{zz}+p_e=3\mu w_z$. Mass conservation and axial force balance therefore give

In steady flow $Aw=Q$. Dividing the momentum equation by $A=Q/w$ givesWith $w=\hat{w}W$ and $z=\hat{z}Z$, chooseThe dimensionless equation becomesTreating $(W^{\prime }{)}^2$ as a function of $W$ and using an integrating factor yieldsBecause $A=Q/(\hat{w}W)$, the area sketches are the reciprocals of the functions $W$ found below. The dimensional vertical deviatoric stress is

The broad faces have curvature zero to leading order, so their stress boundary condition gives ${\sigma }_{xx}=-p_e$. The semicircular edges have curvature $1/h$, giving ${\sigma }_{yy}=-p_e-\gamma /h$. Incompressible flow gives $u_x+v_y+w_z=0$; uniform transverse normal stresses imply uniform transverse extension rates, and eliminating them from the Newtonian fluid stress tensor yields

At an edge, the kinematic boundary condition balances axial advection of $b$, lateral strain, and capillary retraction. This givesThe equation $hbw=Q$ expresses conservation of volume flux, whilestates that the total axial tension is constant. DefineThen $[\Gamma ]=L^{-2}$ and $[T]=L^{-1}$, and substitution of $h=Q/(bw)$ givesSubtracting these logarithmic-derivative equations givesThe remaining width equation isThus, for $T\ne 0$,while the continuous $T=0$ limit is $1/b=1/b_0+\Gamma z/6$.

If $\Gamma =0$, then $b/b_0=e^{-Tz/6}$, $w/w(0)=e^{Tz/3}$, and mass conservation gives $h/h(0)=e^{-Tz/6}$. Therefore a prescribed thinning ratio $R=h(L)/h(0)$ requires

Let $K=k\Delta \rho g/\mu$. Hydrostatic pressure continuity at the lower boundary of the light current givesinside the current, so $p_x=\Delta \rho g{h}_x$. Darcy law gives the depth-integrated horizontal flux per unit transverse widthThus the porous gravity current satisfiesaway from the fracture. The boundary and front conditions areAt $x=L$, the pressure excess at the base of the fracture is $\Delta \rho g{h}_L$. Taking upward leakage as positive,This jump is the local mass conservation law for a leaky porous gravity current.

At late times, $q=Q$ to leading order on $0<x<L$, while $Q_l\simeq Q$. Integrating $q=-K(h^2{)}_x/2$ givesLeakage balance and the pressure-driven drop determine

In the far field, $h(L,t)\simeq h_L$ andBalancing the two sides with $h=O(h_L)$ givesMore precisely, setThe self-similar solution is determined byand $x_N=L+{\eta }_N(K{h}_{L}t/\varphi {)}^{1/2}$.

A shelfy stream is grounded ice with nearly depth-independent horizontal velocity. If the lubricating till has thickness $d$ and viscosity ${\mu }_t$, its simple-shear traction is approximately ${\mu }_{t}u/d$. Comparison with ${\tau }_b=\mu u/\lambda$ givesso $\lambda$ has dimensions of length.

Hydrostatic pressure contributes the depth-integrated longitudinal force $-\rho g{h}^2/2$, while the Newtonian extensional stress contributes $4\mu h{u}_x$ by the shallow-shelf approximation. Balancing the change of their sum against basal drag on a slice givesFor a freely floating ice shelf, hydrostatic flotation reduces the gravitational driving by $1-\rho /{\rho }_w$ and removes basal drag:Depth-integrated mass conservation isin the absence of accumulation or ablation.

In a thin stream with horizontal scale $L_x$, the ratio of extensional resistance to basal drag is $O(\lambda h/L_x^2)$, which is small when the thin-film condition $h/L_x\ll 1$ is combined with $\lambda \ll h_G$. For steady flux $hu=q$, neglecting extension givesAt the grounding line, flotation over a bed depth $b$ givesIntegration yieldsThe grounding-line slope is $|h_x(0)|=\mu q/(\rho g\lambda h_G^2)$. Thin-film theory therefore also requiresThis and $\lambda \ll h_G$ are compatible precisely when $\mu q/(\rho g{h}_G^3)\ll 1$.

The total horizontal force resultant in the stream isThe floating shelf equations and its ocean-front traction imply that the force it exerts at the grounding line is the ocean's hydrostatic force, soUsing flotation and the approximate stream solution,Consequentlyand the grounding-line flux-thickness relation is

At the final state of the slump test for yield stress, an axisymmetric deposit of height $h(r)$ is just able to support itself. Hydrostatic pressure gives radial pressure gradient $\rho g{h}^{\prime }(r)$, while lubrication theory makes the magnitude of the basal shear stressThe marginally yielded final profile therefore satisfiesIntegration givesUsing mass conservation, the known volume isSolving for the yield stress produces the estimateThus one measures the final radius $R$ and inserts it with $V$ and $\rho$. The estimate assumes a thin deposit, negligible surface tension, complete initial yielding, and a spatially uniform yield stress.

Dry sand is a granular material governed primarily by frictional stability. Its final free surface reaches the angle of repose ${\theta }_r$, so the deposit is approximately a cone,This constant-slope profile differs from the square-root edge of the yield-stress-fluid model.

On wavelengths much larger than halos, all matter is partitioned among halos. The mass-weighted halo overdensity must therefore equal the matter overdensity. Since ${\delta }_h(M)=b(M){\delta }_m$, mass conservation requires

For the Press-Schechter formalism, the mass-fraction measure becomesIt is normalized and is a half-normal distribution, soUsing the linear Eulerian halo biastherefore giveswhich verifies the consistency relation.

Since $Df/Dt=0$, the density ansatz hasThe velocity divergence isThe mass conservation equation therefore givesSimilarly,and hence

The $x$ component of the material acceleration isBecause $d\hat{p}/df=\hat{\rho }$,Combining this with $-{\partial }_x\Phi =-{\Omega }^{2}x$ givesThe $y$ and $z$ components give the analogous equations for $b$ and $c$. Finally, $\hat{p}(0)=0$ makes $p=0$ on the material free surface, so both dynamic and kinematic boundary condition requirements are satisfied.

The depth-integrated ice flux isLocal mass conservation with accumulation rate $a$ saysSubstituting the flux givesas required. Equivalently, after using $u_b=-\beta h_x$,