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Along a boundary-layer streamline directed by increasing $s$, an adverse pressure gradient is a rise in pressure in the direction of flow. The associated pressure gradient force opposes the velocity. In a steady inviscid outer stream the Euler equation gives $UdU/ds=-{\rho }^{-1}dp/ds$, so a positive gradient decelerates a positive outer velocity. Within a viscous boundary layer the slower fluid near the wall has less streamwise momentum and can reverse while the outer stream still moves forward. The onset of flow separation is therefore associated with vanishing and then reversed wall shear stress. A pressure rise does not by itself locate separation: the full boundary-layer solution and upstream history also matter.

The geostrophic wind is the horizontal flow for which the pressure gradient force balances the Coriolis force. It runs along atmospheric isobars and has speed proportional to their pressure gradient, at fixed density and nonzero Coriolis parameter. Curvature, friction and small spatial scales can invalidate that approximation.

The gradient wind retains the centripetal acceleration of a curved flow as well as the Coriolis force and horizontal pressure gradient force. For cyclonic circular flow in the northern hemisphere, $v^2/r+fv={\rho }^{-1}dp/dr$, with $r$ increasing away from the low-pressure centre.

In the rotating terrestrial frame, horizontal pressure gradient force and Coriolis force balance in geostrophic balance. Gravity establishes the approximately vertical hydrostatic equilibrium; it is not an additional horizontal force in this balance. With eastward $x$, northward $y$, upward unit vector $\hat{\mathbf{z}}$ and Coriolis parameter $f=2\Omega \sin \varphi$,The geostrophic wind runs along atmospheric isobars, with lower pressure on its left in the Northern Hemisphere.

Interpreting the stated south-to-north gradient as $p_y>0$, $p_y=200\mathrm{Pa}/(2.00\times 10^5\mathrm{m})=10^{-3}\mathrm{Pa}{\mathrm{m}}^{-1}$, while $p_x=0$. Using the external nominal Earth's rotation rate $\Omega =7.292115\times 10^{-5}{\mathrm{s}}^{-1}$ gives the meridional-pressure-gradient geostrophic windBoth zonal velocities are westward; their signs reverse if the supplied gradient is interpreted as a northward decrease. ^[ref]

The validity criteria for geostrophic balance require horizontal acceleration and friction to be small compared with the balancing forces: Rossby number $U/(|f|L)\ll 1$ and slow evolution compared with $1/|f|$. The approximation fails near the equator, on sufficiently small or rapidly changing scales, in strongly curved flow needing appreciable centripetal acceleration, and in the frictional surface layer. Neglecting friction near the ground is the stated idealization.

For an illustrative geostrophic wind with linear drag, use acceleration $-r\mathbf{u}$ with $r>0$. The steady horizontal momentum equation givesDrag reduces the wind and hence its Coriolis acceleration; the wind crosses isobars toward lower pressure. Here it points southwest. The pressure-gradient acceleration points south, Coriolis northwest and drag northeast; the three forces sum to zero. The cross-isobar angle satisfies $\tan \alpha =r/f$ for $f>0$. A turbulent boundary layer has a vertically varying stress rather than necessarily this constant linear drag.

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.

Take $x$ eastward, $y$ northward and $z$ upward. In a rotating reference frame, the horizontal Cauchy momentum equation isHere ${\mathbf{F}}_h$ represents other accelerations, such as friction, and $f$ is the Coriolis parameter. When the Rossby number $U/(|f|L)$ and the ratio $1/(|f|\tau )$ are small, and friction is weak, both advective and local acceleration can be neglected. The Coriolis force then balances the pressure gradient force: this is geostrophic balance, with $fv=p_x/\rho$ and $fu=-p_y/\rho$. The geostrophic wind runs along atmospheric isobars, with low pressure on its left in the northern hemisphere.

A rapid change of pressure need not bring the corresponding geostrophic wind into existence at once. An initially weak wind therefore leaves an inward pressure gradient force around the new low. For a smoothed anomaly of horizontal scale $L=5\times 10^5\mathrm{m}$, geostrophic balance would requireFor the illustrative midlatitude value $|f|=10^{-4}{\mathrm{s}}^{-1}$, the implied Rossby number is already about $0.67$, so curved-flow acceleration matters. With a small tropical value $|f|=10^{-5}{\mathrm{s}}^{-1}$, the inferred speed is $333\mathrm{m}{\mathrm{s}}^{-1}$ and the Rossby number about $67$: the assumed geostrophic balance is untenable. Geostrophic adjustment redistributes the anomaly and emits internal gravity waves, leaving a weaker balanced component.

For a target speed $U_{\ast }=10\mathrm{m}{\mathrm{s}}^{-1}$ on the same scale, the required scale estimate isThus $6\mathrm{hPa}$, or $0.30$ of the initial magnitude, is the midlatitude estimate; at the illustrative tropical $|f|=10^{-5}{\mathrm{s}}^{-1}$ it is $0.6\mathrm{hPa}$, or $0.030$ of the initial magnitude. The paper gives no latitude, so there is no unique numerical reduction. At the equator itself one must use equatorial dynamics rather than divide by $f=0$. Even the tropical $10\mathrm{m}{\mathrm{s}}^{-1}$ estimate has Rossby number $2$, so it describes the scale permitted by the stated balance, rather than a controlled tropical geostrophic balance.

There are two qualifications to “unstable”. A literal top-hat pressure anomaly has a discontinuous edge, where no finite classical geostrophic wind can balance its distributional pressure gradient. The $500\mathrm{km}$ estimate assumes a smoothed edge; a thinner edge requires stronger winds. More generally a large low is not, by its amplitude alone, proved to have an exponentially growing instability. For a concrete counterexample to the claim that such an amplitude cannot be stationary, set $p\prime (s)=-P\exp (-s^2/a^2)$, with cylindrical radius $s$, constant $\rho$ and $f>0$. The cyclonic gradient windis real and satisfies $v_{\theta }^2/s+f{v}_{\theta }=p_s/\rho$ for every $P>0$, with a regular limit at the centre. This is a stationary axisymmetric flow; no claim about its perturbation stability is needed. Here the physically justified conclusion is that a rapidly imposed anomaly without its balancing wind is unbalanced.

Small tropical Coriolis parameter allows only weak horizontal pressure gradients to coexist with ordinary large-scale winds. Hydrostatic balance connects the vertical variation of these pressure gradients to horizontal temperature gradients; equivalently the approximate thermal wind relation gives $f{u}_z\simeq -(g/T)T_y$. For comparable vertical wind shear, small $|f|$ means smaller $|T_y|$. Internal gravity waves can spread heating-induced buoyancy anomalies efficiently before rotation arrests the adjustment. This motivates the weak temperature gradient approximation for the tropical free troposphere; it does not require the land surface or atmospheric boundary layer to have uniform temperature.

The equations describe the linear reduced gravity shallow water equations for a uniform upper layer over a deep, resting lower layer, or equivalently one separated internal vertical mode. The layer is shallow compared with its horizontal scales, permitting hydrostatic balance, and its horizontal velocity is independent of depth. The Boussinesq approximation retains the small density contrast through $g\prime =g({\rho }_2-{\rho }_1)/{\rho }_0>0$. We neglect friction, external forcing, background currents, and quadratic perturbation terms. On an equatorial beta plane, $f=\beta y$ with $\beta >0$, and the equatorial Kelvin wave branch imposes $v=0$. The meridional Cauchy momentum equation then reduces exactly within this linear model to a balance between the Coriolis force and pressure gradient force.

Strictly, the printed $h$ must denote the thickness perturbation $\eta$, with total thickness $h_0+\eta$ and $|\eta |\ll h_0$. A signed oscillation around zero cannot represent the total positive layer thickness. Define $c=\sqrt{g\prime h_0}$. For a nonzero zonal wavenumber $k$, the zonal Cauchy momentum equation and conservation of mass giveOn the eastward branch $\omega =ck$, the wave polarization is $u=(c/h_0)\eta$. Meridional geostrophic balance givesThe branch $\omega =-ck$ instead grows as $\exp (+\beta y^2/(2c))$ and is excluded by decay away from the equator. Matching the paper's Gaussian function therefore givesThe usual equatorial deformation radius is $R_{\mathrm{eq}}=\sqrt{c/\beta }$, with the envelope written $e^{-y^2/(2R_{\mathrm{eq}}^2)}$. Thus the printed exponent uses $L_{\mathrm{eq}}=\sqrt{2}{R}_{\mathrm{eq}}$. Calling that width a deformation radius is a convention; using the standard radius without the factor $1/2$ would fail the printed meridional equation.

At an instant, alternating positive and negative thickness anomalies have their largest magnitudes on the equator and decay symmetrically away from it. Eastward arrows occupy the positive anomalies, westward arrows the negative anomalies; the velocity vanishes at the nodes and weakens away from the equator. The whole pattern travels eastward at speed $c$. Meridionally, $fu=-g\prime {\eta }_y$ is geostrophic balance. For a positive anomaly north of the equator, the northward pressure gradient force balances the southward Coriolis force; both directions reverse south of the equator. Zonally, $fv=0$ and $u_t=-g\prime {\eta }_x$: the zonal pressure gradient force accelerates the flow, so it is not in zonal geostrophic balance. On the equator the meridional terms both vanish smoothly.

Use eastward $x$, northward $y$, constant layer depth $H$, constant reference density ${\rho }_0$, and a depth-independent horizontal velocity field $(u,v)$. Assume a flat bottom and a rigid upper boundary, giving the two-dimensional incompressibility condition $u_x+v_y=0$. The traditional approximation retains only the vertical component of planetary rotation in the horizontal equations. Take the Coriolis parameter $f=f(y)$, a constant linear drag rate $r$, and no other torques. The component Cauchy momentum equations areDefine relative vorticity $\xi =v_x-u_y$. Apply ${\partial }_x$ to the second equation and subtract ${\partial }_y$ of the first. The pressure gradient force has zero curl because ${\rho }_0$ is constant and mixed partial derivatives commute. Direct expansion of the material-acceleration terms givesThe curl of the rotation terms isThe imposed zonal wind stress contributes $-{\tau }_{s,y}/({\rho }_{0}H)$, and constant linear drag contributes $-r\xi$. HenceSince $Df=v{f}_y$, defining absolute vorticity $\zeta =\xi +f$ givesWithout the incompressibility condition, the left-hand side would contain the stretching term $\zeta {\nabla }_h\cdot \mathbf{u}$. Variable layer depth, variable density or spatially varying linear drag would introduce further terms. No small-Rossby number assumption was needed to obtain this constant-depth vorticity equation.

Take $x$ eastward, $y$ northward and $z$ upward, and let $u$ be the eastward geostrophic wind. With Coriolis parameter $f$ independent of $z$, geostrophic balance, hydrostatic balance, and the ideal gas law giveHere $R_d$ is the specific gas constant for dry air. Taking a vertical derivative of the first equation and interchanging mixed partial derivatives givesIt matters that $\rho$ is not constant in the pressure gradient force. Substitutingcancels the two terms proportional to $gfu/(R_{d}T)$ and yields the height-coordinate thermal wind relationIf $|u{T}_z|\ll |T{u}_z|$, this reduces toThe discarded term has relative scale $H|T_z|/T$ when $u_z\sim u/H$. Taking $H=10\mathrm{km}$, $T=250\mathrm{K}$ and a atmospheric lapse rate of $6.5\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$ gives about $0.26$. Thus this height-coordinate formula is a useful estimate, with potentially tens-of-percent compressibility-related correction through the vertical temperature gradient; it is not an exact consequence of the two balances. If the temperature gradient vanishes vertically, the displayed approximation is exact within those balances.

The hydrostatic approximation is very good on the broad, shallow atmospheric scales considered. The geostrophic balance is normally justified by a small Rossby number and fails as a general approximation near the equator. At $30^{\circ }\mathrm{N}$, $f=7.292\times 10^{-5}{\mathrm{s}}^{-1}$. With the ideal $134\mathrm{m}{\mathrm{s}}^{-1}$ wind and a $1000$--$2000\mathrm{km}$ horizontal scale, $\mathrm{Ro}=U/(fL)\simeq 0.9$--$1.8$: quantitative accuracy is limited for this especially strong ideal jet. Small vertical aspect ratio alone does not ensure geostrophic balance, although a straight, steady zonal flow can satisfy it even when this scale estimate of the Rossby number is not small.

For the requested estimate, suppose the upper jet stream is $H=10\mathrm{km}$ above a weak surface wind, has temperature $T\simeq 250\mathrm{K}$, and spans a meridional width $L$. Integrating the approximate thermal wind relation in scale form givesThe northward temperature gradient is negative because the eastward wind strengthens upward. A $1000$--$2000\mathrm{km}$ jet width gives roughly $25$--$50\mathrm{K}$. No unique numerical difference follows without specifying the jet width, height and representative temperature; these are explicitly estimated quantities.

A horizontal pressure gradient force accelerates air from higher toward lower pressure. In a rotating frame the Coriolis force deflects the flow. Away from strong friction, at large spatial scales and small Rossby number, a slowly changing flow can reach geostrophic balance:where $f=2\Omega \sin \phi$ is the Coriolis parameter. The geostrophic wind runs along atmospheric isobars, with speed $|{\nabla }_{h}p|/(\rho |f|)$. Closely spaced atmospheric isobars therefore indicate stronger geostrophic wind when density and latitude are comparable. In the northern hemisphere lower pressure lies to the left of this wind, producing counterclockwise circulation around a low; the sense reverses in the southern hemisphere.

A curved flow also has centripetal acceleration. For cyclonic circular flow in the northern hemisphere, the gradient wind relation isNear the ground, friction reduces wind speed and weakens its Coriolis contribution, so air crosses atmospheric isobars toward lower pressure. The simple geostrophic formula fails near the equator, where $f$ becomes small, and where rapid changes or tight curvature make neglected acceleration important. Vertically, large-scale motion is often approximated by hydrostatic balance, $dp/dz=-\rho g$.

Bernoulli's principle can relate speed changes to pressure changes along an appropriate inviscid streamline, but its simple dynamic-pressure relation is not a replacement for the rotating force balance governing a broad atmospheric pressure map.

In the horizontal rotating-fluid equation of motion, scale advective acceleration as $U^2/L$ and Coriolis acceleration as $|f|U$, where $f=2\Omega \sin \varphi$. Their ratio is the Rossby number,A small Rossby number means that rotation strongly constrains the flow; with slow evolution, weak friction and a suitable pressure field it permits geostrophic balance. Local time dependence introduces the separate condition $1/(|f|t_{\mathrm{evol}})\ll 1$. A synoptic midlatitude cyclone with $U\sim 10\mathrm{m}{\mathrm{s}}^{-1}$, $L\sim 1000\mathrm{km}$ and $|f|\sim 10^{-4}{\mathrm{s}}^{-1}$ has $\mathrm{Ro}\sim 0.1$. A tornado with $U\sim 50\mathrm{m}{\mathrm{s}}^{-1}$ and $L\sim 1\mathrm{km}$ has $\mathrm{Ro}\sim 500$; its rotational motion does not mean that the Earth's rotation dominates its local dynamics.

The vertical relative vorticity is $\zeta ={\partial }_{x}v-{\partial }_{y}u\sim U/L$, while planetary vorticity is $f$. Thus $|\zeta /f|$ is of the order of the Rossby number, although it is not an identity for arbitrary velocity fields: an irrotational strain can have nonzero $U/L$ but zero $\zeta$. Absolute vorticity is $f+\zeta$.

On large, slowly evolving midlatitude scales, the horizontal pressure gradient force is mainly balanced by the Coriolis force, which is perpendicular to velocity. Consequently the wind is nearly parallel to atmospheric isobars, rather than accelerating directly into a low or out of a high. Curvature adds a gradient wind correction, and surface friction permits the cross-isobar component considered in part (d).

The pressure gradient is the vector $\nabla p$, pointing in the direction in which pressure increases fastest. Its magnitude is pressure change per unit distance, with units $\mathrm{Pa}{\mathrm{m}}^{-1}$. Across a small fluid element, larger pressure on one face than its opposite produces a net pressure gradient force per unit volume $-\nabla p$.

For $p(x)=p_0-Gx$ with $G>0$, the pressure force points along positive $x$. In a long pipe, it can balance viscous resistance and sustain steady flow. In a stationary vertical column, it instead balances gravity, so a nonzero pressure gradient does not by itself prove that fluid must move. The Cauchy momentum equation determines acceleration from all the forces together. Comparing pressure measurements at separated positions estimates the mean gradient over their spacing.