Ch 5. Geostrophic & hydrostatic balance

Overview

Two balances — geostrophic (Coriolis against pressure gradient) and hydrostatic (gravity against vertical pressure gradient) — together with the chain rule produce one of the most useful diagnostic relations in the subject: thermal wind. It says the vertical shear of the geostrophic wind is fixed entirely by the horizontal temperature gradient, with no need to know the wind itself. This is how the jet stream can be predicted from a weather map showing only temperature, and it’s why the jet sits where it does: exactly where the temperature gradient reverses sign, at the tropopause.

The model

Notebook: notebooks/ch05_geostrophic-balance.py → exported to /nb/ch05_geostrophic-balance.html. Governing equations use the symbols in notation:

u_g=-\frac{\phi_y}{f},\qquad v_g=\frac{\phi_x}{f},\qquad \phi_z=b \quad\Longrightarrow\quad fu_{g,z}=-b_y.

Unlike every other notebook in this book, there is no PDE and no time integration here — gfdlib.balance provides closed-form expressions for an idealized frontal buoyancy field and the geostrophic wind it implies, exact to floating-point precision. Move a slider and the reconstruction updates live.

What to look for

Tighten the front. Shrink $L_y$ at fixed $\Delta b$ : the total temperature contrast across the domain hasn’t changed, but the jet strengthens roughly as $1/L_y$ . Thermal wind responds to the gradient $b_y$ , not the contrast — which is why the jet stream lives over the frontal zone and not over the (much larger) pole-to-equator temperature difference as a whole.
Weaken rotation. Lower $f$ and the same buoyancy field implies a stronger shear ( $u_{g,z}=-b_y/f$ ). Watch the Rossby number in the jet-core readout climb as you do — the diagnostic is telling you it is sawing off the branch it sits on, since geostrophy itself assumes $Ro\ll1$ . This is also why the “dynamic method” fails near the equator.
Move the tropopause. Slide $H_{trop}$ up and down: the jet core rides exactly at the level where $b_y$ changes sign, because that is where the integrand of the thermal-wind integral reverses and the accumulated shear peaks. Nothing about the wind was prescribed at that height — it is all inherited from the mass field.
Check the reversal. The frontal $b_y$ changes sign at $z=H_{trop}$ by construction, so $u_{g,z}=-b_y/f$ must reverse there too: the wind grows monotonically with height through the whole “troposphere,” peaks exactly at the tropopause, and decays above. No extremum-seeking was coded anywhere — the jet maximum is the integral of $-b_y/f$ turning around, which is why real jet cores hug the tropopause on every observed cross-section.

Prefer to run it yourself? Download as a Jupyter notebook — it installs its own dependencies in the first cell. The sliders are static in Jupyter; edit value= and re-run to change a parameter.

Both fluids

Atmosphere: this is literally how forecasters read a thickness chart — tightly packed isotherms on a temperature map mean strong vertical wind shear is guaranteed nearby, jet stream included, before a single wind observation is consulted.
Ocean: the same relation underlies the dynamic method used for a century to estimate ocean currents from hydrographic (temperature/salinity) sections alone — geostrophic currents relative to a reference level are computed directly from the density field, exactly as this notebook reconstructs $u_g$ from $b$ .

Exercises

(analytic) Derive $fu_{g,z}=-b_y$ from geostrophic and hydrostatic balance, starting from the horizontal momentum and hydrostatic equations. Show that at low Rossby number the acceleration terms are the ones that drop out, not the Coriolis or pressure-gradient terms.
(computational) Modify ch05_geostrophic-balance.py to use a double-front buoyancy field (two tanh fronts of opposite sign, representing a polar front and a subtropical front). How many jet cores appear, and where?
(exploratory) At what value of $Ro=U_{jet}/(fL_y)$ (read from the jet-core plot’s title) would you say geostrophic balance has stopped being a good approximation? Push $f$ down until you’re not sure anymore.