Hydrostatic Equilibrium Questions and Answers

Answer: C

Integrating the hydrostatic equation with the gas law substituted gives the geopotential thickness as proportional to the mean virtual temperature of the layer times the logarithm of the pressure ratio. That is the hypsometric equation. The hydrostatic equation and the gas law are its ingredients, and the thermal wind relation is obtained by differentiating it horizontally. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

Answer: D

The mean is taken with respect to the logarithm of pressure, and virtual temperature is used so that the dry air gas constant can be retained. In practice the difference between an arithmetic and a logarithmic mean is small for a thin layer. Using the base temperature or a midpoint value would be an approximation to the correct mass-weighted mean. Framing the question as which quantity the change acts upon makes the answer follow without calculation.

Answer: A

Because the only variable in the hypsometric relation for a fixed pair of pressure levels is the layer mean temperature, contours of thickness are contours of mean temperature. This is why thickness charts are used for air mass identification and for forecasting precipitation type. Humidity enters only weakly through virtual temperature. Checking the units of each quantity catches a wrong formula before the arithmetic begins.

Answer: B

The 1000 to 500 hectopascal thickness spans roughly the lower half of the atmosphere by mass and is the traditional field for air mass analysis. Typical mid-latitude values run from about 510 decametres in very cold air to 570 in warm tropical air. The other layers are used for more specialised diagnostics.

Answer: C

The 540 decametre contour of the 1000 to 500 hectopascal thickness is the classical rule of thumb for the boundary between rain and snow at low levels. It is only a guide, since the distribution of temperature within the layer matters as much as its mean. The 528 value is genuinely used as a stricter criterion, which is why it makes a plausible alternative. A limit that survives making every external component ideal is an intrinsic one, and worth knowing for that reason.

Answer: D

Thickness is a mass-weighted mean, so it conceals the vertical distribution, and a warm nose only a few hundred metres deep can turn snow into rain or freezing rain. Examining the full profile is always preferable. The computation over high ground does involve a fictitious layer, which makes that a genuine secondary difficulty rather than a wholly wrong answer. The shared properties explain why these belong to one family, and the differences explain why more than one exists.

Answer: A

Geopotential is the potential energy per unit mass relative to sea level, and dividing it by a standard gravity gives geopotential height. Using it absorbs the variation of gravity with latitude and altitude into the coordinate, so that a given geopotential height corresponds to the same energy everywhere. Pressure and temperature are different variables entirely. Framing the question as which quantity the change acts upon makes the answer follow without calculation.

Answer: B

Because geopotential height is defined through the work done against gravity, surfaces of constant geopotential height are surfaces of constant energy, and the variation of gravity no longer complicates the equations. Radiosondes measure pressure and temperature and derive height hypsometrically, so temperature is very much required. The two heights differ by only a small amount that changes sign with latitude. The quantity that varies and the quantity that does not are worth naming before any substitution is attempted.

Answer: C

The two differ by a few tens of metres at 10 kilometres, with geopotential height smaller at low latitudes where gravity is weaker and larger at high latitudes. The difference is small enough to ignore for most purposes but large enough to matter in precise work. Being aware that a difference exists is more important than remembering its exact magnitude. The quantity that varies and the quantity that does not are worth naming before any substitution is attempted.

Answer: D

In pressure coordinates the horizontal pressure gradient force becomes minus the gradient of geopotential along the pressure surface, with density having vanished entirely. Continuity also becomes simpler, losing its density and its time derivative. Gravity is absorbed into the geopotential rather than disappearing, and vertical velocity remains unmeasurable. Reading the definition literally, rather than recalling a formula, is the reliable route through questions of this shape.