Thermal Wind Balance Questions and Answers

Q: Which mechanism does the Eady solution describe as producing growth?

Each boundary supports a wave that propagates against the local flow, and at the most unstable wavelength the two lock into a fixed phase relationship with a westward tilt and amplify each other. Growth therefore requires both boundaries. Latent heat is absent from the dry model although it matters greatly in reality, friction is neglected entirely, and gravity wave reflection describes a quite different class of solution that carries no available potential energy.

Q: Which quantity must change sign somewhere for baroclinic instability to be possible?

The Charney Stern criterion requires the meridional potential vorticity gradient to change sign somewhere in the domain, with boundary temperature gradients counting as delta functions in that gradient. In the Eady model the interior gradient is zero and the two boundaries supply the required change of sign. Constant shear, a fixed Coriolis parameter and positive static stability are all perfectly compatible with instability, so none of the other conditions is necessary.

Q: Which realistic feature does the Charney model add to the Eady model?

Charney worked on a beta plane with no upper lid, so the interior potential vorticity gradient is non-zero and instability arises from its interaction with the surface temperature gradient. The solutions are bottom-trapped with a depth set by the Rossby height. Latent heat, friction and realistic stability profiles all matter in the real atmosphere but are absent from both idealised models, which is what keeps them tractable and instructive.

Q: Which region shows the largest values of the Eady growth rate parameter?

Strong shear and modest stratification combine off the eastern coasts of Asia and North America in winter, which is exactly where the observed storm tracks begin. The agreement between the theoretical parameter and the observed climatology is striking. The equatorial belt has almost no Coriolis parameter, deserts have weak vertical shear despite hot surfaces, and continental inversions represent strong stratification that suppresses rather than encourages growth.

Q: Which effect does strong static stability have on baroclinic growth?

Growth rate is inversely proportional to the buoyancy frequency, so a strongly stratified atmosphere resists the vertical displacement that the conversion of potential to kinetic energy requires. This is why the stably stratified stratosphere does not support baroclinic cyclones. Stability does also lengthen the most unstable wavelength through the Rossby height, so the final option has the direction of that effect the wrong way round as well.

Q: Which scale does the ratio of the Coriolis parameter to the buoyancy frequency define?

The ratio \( f/N \) is about one in a hundred in the troposphere, and multiplying it by a horizontal scale gives the depth over which a disturbance of that width can influence the flow. It is therefore the natural aspect ratio of balanced motion and explains why weather systems a thousand kilometres wide are about ten kilometres deep. The Rossby, Richardson and Burger numbers are formed from different combinations of the same quantities. Naming the cost of a choice alongside its benefit is what turns a fact into a design decision.

Q: Which depth does a disturbance of 1000 kilometres horizontal scale influence?

Multiplying the horizontal scale by \( f/N \), about one in a hundred, gives a Rossby height near 10 kilometres, which is the depth of the troposphere. This is why synoptic systems are effectively deep, occupying the whole troposphere, while a disturbance of only a hundred kilometres width would influence just a kilometre of depth. The estimate explains a great deal about the observed structure of weather systems in one short calculation. A single measurement chosen to respond to the one asymmetry beats several that respond to shared behaviour.

Q: Which conclusion follows about the depth of a small-scale vortex?

The Rossby height is proportional to the horizontal scale, so a vortex of 100 kilometres width influences only about a kilometre of depth. This is why mesoscale vortices are shallow features while synoptic cyclones are deep ones. The result also explains why an upper level potential vorticity anomaly must be reasonably broad if it is to induce any appreciable circulation at the surface far below it.

81.

Which mechanism does the Eady solution describe as producing growth?

A.Two edge waves phase-locking against the shear
B.The release of latent heat as saturated air ascends within the warm sector of the developing wave
C.Frictional convergence
D.The reflection of gravity waves between the two rigid boundaries bounding the flow above and below

Answer: A

Each boundary supports a wave that propagates against the local flow, and at the most unstable wavelength the two lock into a fixed phase relationship with a westward tilt and amplify each other. Growth therefore requires both boundaries. Latent heat is absent from the dry model although it matters greatly in reality, friction is neglected entirely, and gravity wave reflection describes a quite different class of solution that carries no available potential energy.

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82.

Which quantity must change sign somewhere for baroclinic instability to be possible?

A.The static stability, which must become negative within a layer for a disturbance to grow at all
B.The meridional gradient of potential vorticity
C.The vertical shear of the geostrophic wind, which must reverse at some level within the flow domain
D.The Coriolis parameter

Answer: B

The Charney Stern criterion requires the meridional potential vorticity gradient to change sign somewhere in the domain, with boundary temperature gradients counting as delta functions in that gradient. In the Eady model the interior gradient is zero and the two boundaries supply the required change of sign. Constant shear, a fixed Coriolis parameter and positive static stability are all perfectly compatible with instability, so none of the other conditions is necessary.

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83.

Which role do the boundary temperature gradients play in the stability criterion?

A.They set the growth rate
B.They fix the wavelength at which the disturbance grows most rapidly within the available spectrum
C.They act as concentrated potential vorticity gradients
D.They determine the speed at which the growing disturbance propagates along the boundary concerned

Answer: C

A temperature gradient at a rigid boundary is dynamically equivalent to a sheet of potential vorticity gradient concentrated at that surface, which is how the Eady model satisfies the Charney-Stern criterion despite having no interior gradient at all. This equivalence, due to Bretherton, unifies the treatment of boundary and interior effects. Propagation speed, growth rate and preferred wavelength all follow from the full solution rather than from the boundary gradients alone.

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84.

Which realistic feature does the Charney model add to the Eady model?

A.The release of latent heat within the ascending branch of the growing baroclinic wave being modelled
B.Surface friction
C.A realistic vertical profile of static stability including the sharp increase at the tropopause
D.The variation of the Coriolis parameter with latitude

Answer: D

Charney worked on a beta plane with no upper lid, so the interior potential vorticity gradient is non-zero and instability arises from its interaction with the surface temperature gradient. The solutions are bottom-trapped with a depth set by the Rossby height. Latent heat, friction and realistic stability profiles all matter in the real atmosphere but are absent from both idealised models, which is what keeps them tractable and instructive.

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85.

Which practical measure of baroclinicity combines shear and stratification?

A.The Eady growth rate parameter
B.The bulk Richardson number, which is used to assess the likelihood of turbulence within a sheared layer
C.The thickness gradient
D.The Brunt-Vaisala frequency, which measures the frequency at which a displaced parcel will oscillate

Answer: A

The Eady growth rate parameter, proportional to the Coriolis parameter times the vertical shear divided by the buoyancy frequency, is widely used to map where the atmosphere is most prone to baroclinic development. It combines exactly the ingredients the theory identifies. The bulk Richardson number addresses small-scale turbulence, the thickness gradient captures only the shear part, and the buoyancy frequency only the stratification. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

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86.

Which region shows the largest values of the Eady growth rate parameter?

A.The interior of the winter continents, where surface cooling produces the strongest surface inversions
B.The western oceanic storm tracks in winter
C.The subtropical deserts in summer, where the surface temperature gradient reaches its greatest values
D.The equatorial belt

Answer: B

Strong shear and modest stratification combine off the eastern coasts of Asia and North America in winter, which is exactly where the observed storm tracks begin. The agreement between the theoretical parameter and the observed climatology is striking. The equatorial belt has almost no Coriolis parameter, deserts have weak vertical shear despite hot surfaces, and continental inversions represent strong stratification that suppresses rather than encourages growth.

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87.

Which effect does strong static stability have on baroclinic growth?

A.It has no effect
B.It shifts the most unstable wavelength to shorter values without changing the rate at which it grows
C.It suppresses growth
D.It enhances growth because a strongly stratified atmosphere stores more available potential energy

Answer: C

Growth rate is inversely proportional to the buoyancy frequency, so a strongly stratified atmosphere resists the vertical displacement that the conversion of potential to kinetic energy requires. This is why the stably stratified stratosphere does not support baroclinic cyclones. Stability does also lengthen the most unstable wavelength through the Rossby height, so the final option has the direction of that effect the wrong way round as well.

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88.

Which scale does the ratio of the Coriolis parameter to the buoyancy frequency define?

A.The Rossby number of the flow, which compares the inertial and rotational accelerations acting on it
B.The Richardson number
C.The Burger number, which compares the deformation radius against the horizontal scale of the motion
D.The aspect ratio of balanced motion

Answer: D

The ratio \( f/N \) is about one in a hundred in the troposphere, and multiplying it by a horizontal scale gives the depth over which a disturbance of that width can influence the flow. It is therefore the natural aspect ratio of balanced motion and explains why weather systems a thousand kilometres wide are about ten kilometres deep. The Rossby, Richardson and Burger numbers are formed from different combinations of the same quantities. Naming the cost of a choice alongside its benefit is what turns a fact into a design decision.

Discuss in Forum

89.

Which depth does a disturbance of 1000 kilometres horizontal scale influence?

A.About 10 kilometres
B.About 1 kilometre, which is comparable with the depth of the frictional boundary layer over land
C.About 100 kilometres
D.About 3 kilometres, which is roughly the height at which the 700 hectopascal surface is usually found

Answer: A

Multiplying the horizontal scale by \( f/N \), about one in a hundred, gives a Rossby height near 10 kilometres, which is the depth of the troposphere. This is why synoptic systems are effectively deep, occupying the whole troposphere, while a disturbance of only a hundred kilometres width would influence just a kilometre of depth. The estimate explains a great deal about the observed structure of weather systems in one short calculation. A single measurement chosen to respond to the one asymmetry beats several that respond to shared behaviour.

Discuss in Forum

90.

Which conclusion follows about the depth of a small-scale vortex?

A.It penetrates further than a large vortex because its stronger rotation overcomes the stratification
B.Its influence is confined to a shallow layer
C.It extends through the whole troposphere in the same way that a synoptic scale system does normally
D.It has no vertical structure

Answer: B

The Rossby height is proportional to the horizontal scale, so a vortex of 100 kilometres width influences only about a kilometre of depth. This is why mesoscale vortices are shallow features while synoptic cyclones are deep ones. The result also explains why an upper level potential vorticity anomaly must be reasonably broad if it is to induce any appreciable circulation at the surface far below it.

Discuss in Forum

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