Hydrostatic Equilibrium Questions and Answers

Answer: A

A quantity can replace height as a coordinate only if it varies monotonically along the vertical, and hydrostatic balance with positive density guarantees exactly that for pressure. Isothermality, geostrophy and a particular temperature gradient are all irrelevant to the requirement. The monotonic decrease is why hydrostatic balance is a prerequisite for the whole pressure coordinate framework. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

Answer: B

Sigma is pressure divided by surface pressure, so it equals one at the ground whatever the orography and zero at the model top. This removes the problem of pressure surfaces intersecting mountains. The price is that the horizontal pressure gradient becomes a difference of two large terms over steep terrain, which is a well known source of numerical error. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

Answer: C

On a steeply sloping sigma surface the horizontal pressure gradient is computed as the difference between two terms each much larger than their difference, so truncation error can produce a spurious wind. Hybrid coordinates that become pressure surfaces aloft were developed partly for this reason. Mass conservation and the diagnosis of vertical motion are unaffected. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

Answer: D

The hybrid sigma-pressure coordinate follows the terrain near the ground, where that is necessary, and relaxes to pure pressure surfaces in the upper troposphere and stratosphere where the terrain is irrelevant. This captures the advantages of both. Pure isentropic coordinates have attractions for tracer transport but present difficulties where the stratification becomes neutral. The shared properties explain why these belong to one family, and the differences explain why more than one exists.

Answer: A

Integrating the hydrostatic equation with an assumed temperature profile gives pressure as a function of height, which is the barometric formula. The isothermal case gives an exponential and a constant lapse rate gives a power law. Altimeters are calibrated using the constant lapse rate form of the standard atmosphere.

Answer: B

With temperature falling linearly with height, integration gives pressure proportional to the temperature raised to the power \( g/(R\Gamma) \), where \( \Gamma \) is the lapse rate. For the standard value of 6.5 kelvin per kilometre that exponent is about 5.26. The exponential form applies only to the isothermal case, which is a good approximation in the lower stratosphere. A limit that survives making every external component ideal is an intrinsic one, and worth knowing for that reason.

Answer: C

The 500 hectopascal surface lies near 5.5 kilometres in the standard atmosphere, with half the atmospheric mass beneath it. The 700 hectopascal surface is near 3 kilometres and the 850 near 1.5. Memorising the approximate heights of the standard analysis levels makes it far easier to interpret a chart without consulting a table. Naming the cost of a choice alongside its benefit is what turns a fact into a design decision.

Answer: D

The 850 hectopascal surface lies near 1.5 kilometres, just above the top of the boundary layer over most lowland areas, which is why it is used for low level thermal advection. The approximate heights of 1.5, 3, 5.5, 9 and 12 kilometres for the 850, 700, 500, 300 and 200 surfaces form a useful ladder to carry in mind when reading upper air charts. The shared properties explain why these belong to one family, and the differences explain why more than one exists.

Answer: A

Cold air has a smaller scale height, so the pressure falls more rapidly with height and a given pressure surface is reached at a lower altitude. This is why upper level troughs correspond to cold air and ridges to warm air. The relationship between the thermal and the height field is the whole content of the thickness concept. A limit that survives making every external component ideal is an intrinsic one, and worth knowing for that reason.

Answer: B

Because thickness is proportional to mean temperature, a column of cold air has a lower geopotential height at any upper level, appearing as a trough. This is why upper level charts and thickness charts show such similar patterns. The correspondence strengthens with height as the thickness anomaly accumulates through more of the atmosphere. A limit that survives making every external component ideal is an intrinsic one, and worth knowing for that reason.