Divergence and Vertical Motion Questions and Answers

Q: The zonal gradient of meridional current and the meridional gradient of zonal current is \(-0.3 \times 10^{-3}\) s-1 and \(0.3 \times 10^{-3}\) s-1, respectively, at a location P (87 °E, 15 °N). Which one of the following best explains the nature of the flow?

The flow is clockwise in nature.The vertical component of relative vorticity is the zonal gradient of the meridional current minus the meridional gradient of the zonal current:\[ \begin{aligned} \zeta &= \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \\ &= (-0.3 - 0.3) \times 10^{-3} = -0.6 \times 10^{-3}\ \text{s}^{-1} \end{aligned} \]Negative vorticity means clockwise rotation; at 15 °N this is anticyclonic flow.The flow is clearly rotational, since \(\zeta \ne 0\). Counter-clockwise rotation would need positive vorticity. Divergence needs the gradients \(\partial u/\partial x\) and \(\partial v/\partial y\), which are not given, so nothing can be said about it from these data.

Q: Which of the following is the correct form of the mass divergence form of the continuity equation for a compressible fluid?[In the given equations, \(\rho\) is the density and v the three dimensional velocity vector of the fluid.](i) \(\frac{\partial \rho}{\partial t} + \nabla \times (\rho \mathbf{v}) = 0\)(ii) \(\frac{\partial \rho}{\partial t} + \nabla . (\rho \mathbf{v}) = 0\)(iii) \(\frac{\partial \mathbf{v}}{\partial t} + \rho . \nabla \mathbf{v} = 0\)(iv) \(\frac{\partial \rho}{\partial t} + \mathbf{v} . \nabla \rho = 0\)

The mass divergence form is equation (ii) only.Conservation of mass for a compressible fluid states that the local rate of change of density is balanced by the divergence of the mass flux:\(\dfrac{\partial \rho}{\partial t} + \nabla \cdot (\rho\mathbf{v}) = 0\)Equation (i) uses the curl instead of the divergence, and a curl is a vector while \(\partial\rho/\partial t\) is a scalar, so it cannot be a mass balance. Equation (iii) describes the time change of velocity, not of density. Equation (iv) says that density does not change following a fluid parcel, \(D\rho/Dt = 0\), which is the condition for incompressible flow; the full advective form of continuity is \(D\rho/Dt + \rho\nabla \cdot \mathbf{v} = 0\).

Q: In regions when there is a net outflow of air, a situation referred to as :

A net outflow of air is called Divergence.Where more air leaves a region than enters it, the air is diverging. At the surface this happens under high pressure, where sinking air spreads out; in the upper troposphere divergence above a low draws air up from below.Convergence is the opposite, a net inflow of air. Advection is horizontal transport of a property such as heat or moisture by the wind, and convection is vertical transfer of heat by rising and sinking air, so neither names a net outflow.

Q: Which quantity does horizontal divergence measure?

Divergence is the fractional rate of increase of the horizontal area of a fluid element, so positive values mean the element is spreading out and negative values mean it is contracting. Rotation about a vertical axis is vorticity, circulation is the loop integral of velocity, and area-preserving distortion is deformation. Together these four quantities describe every way in which the local velocity gradient can affect a fluid element, and separating them cleanly is the foundation of kinematic analysis in this subject.

Q: Which physical law connects horizontal divergence to vertical motion?

Mass continuity requires that air converging horizontally into a column must leave it vertically, so convergence at low levels implies ascent above. This is the single most used consequence of continuity in synoptic meteorology and it is why surface lows are cloudy and surface highs are clear. Momentum conservation is the equation of motion, energy conservation is the thermodynamic equation, and potential vorticity conservation is a derived result that follows from several of the governing equations combined.

Q: Which form does the continuity equation take in pressure coordinates?

In pressure coordinates continuity becomes a purely diagnostic relation with no density and no time derivative, stating that the three-dimensional divergence vanishes. This is one of the chief practical advantages of the coordinate, because it makes omega directly computable by integrating the horizontal divergence in the vertical. The other options give the height coordinate form, a general statement of mass conservation, and the surface pressure tendency equation respectively.

Q: Which quantity is omega in pressure coordinates?

Omega is the rate of change of pressure following the parcel, so it is the pressure coordinate analogue of the vertical velocity. Because pressure decreases upward, omega is negative for ascent, a sign convention that catches many people out. The height tendency of a pressure surface is a local rather than a material derivative, and the rate of change of potential temperature is the diabatic heating, which is a thermodynamic quantity. Framing the question as which quantity the change acts upon makes the answer follow without calculation.

Q: Which sign does omega take when air is rising?

Pressure decreases with height, so a rising parcel experiences falling pressure and omega is negative. Model output charts therefore show ascent, and hence cloud and precipitation, as negative values. The convention follows entirely from the coordinate being inverted relative to height, and remembering it as a consequence rather than as an isolated fact makes it much harder to get wrong when reading a chart in a hurry. The shared properties explain why these belong to one family, and the differences explain why more than one exists.

Q: Which relationship connects omega to the geometric vertical velocity \( w \)?

Applying the hydrostatic relation to the material derivative of pressure gives \( \omega \approx -\rho g w \), where the small horizontal advection of pressure has been neglected. With a density near 1 kilogram per cubic metre near the surface, a vertical velocity of 1 centimetre per second corresponds to an omega of about 0.1 pascals per second. Keeping this conversion in mind allows model omega fields to be interpreted in physically familiar terms.

Q: Which typical magnitude does synoptic scale omega have?

A vertical velocity of about 1 centimetre per second corresponds to an omega of roughly 0.1 pascals per second, so synoptic values are a few tenths of a pascal per second at most. Convective updraughts of several metres per second give values a hundred times larger. Knowing the characteristic magnitude allows an implausible model field to be spotted immediately, which is a practical skill worth having. Framing the question as which quantity the change acts upon makes the answer follow without calculation.

The zonal gradient of meridional current and the meridional gradient of zonal current is \(-0.3 \times 10^{-3}\) s^-1 and \(0.3 \times 10^{-3}\) s^-1, respectively, at a location P (87 °E, 15 °N). Which one of the following best explains the nature of the flow?

A.The flow is non-divergent in nature.
B.The flow is non-rotational in nature.
C.The flow is counter-clockwise in nature.
D.The flow is clockwise in nature.

Exam Year(s)

GATE XE - 2025,

Answer: D

The flow is clockwise in nature.

The vertical component of relative vorticity is the zonal gradient of the meridional current minus the meridional gradient of the zonal current:

\[ \begin{aligned} \zeta &= \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \\ &= (-0.3 - 0.3) \times 10^{-3} = -0.6 \times 10^{-3}\ \text{s}^{-1} \end{aligned} \]

Negative vorticity means clockwise rotation; at 15 °N this is anticyclonic flow.

The flow is clearly rotational, since \(\zeta \ne 0\). Counter-clockwise rotation would need positive vorticity. Divergence needs the gradients \(\partial u/\partial x\) and \(\partial v/\partial y\), which are not given, so nothing can be said about it from these data.

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Which of the following is the correct form of the mass divergence form of the continuity equation for a compressible fluid?

[In the given equations, \(\rho\) is the density and v the three dimensional velocity vector of the fluid.]

(i) \(\frac{\partial \rho}{\partial t} + \nabla \times (\rho \mathbf{v}) = 0\)
(ii) \(\frac{\partial \rho}{\partial t} + \nabla . (\rho \mathbf{v}) = 0\)
(iii) \(\frac{\partial \mathbf{v}}{\partial t} + \rho . \nabla \mathbf{v} = 0\)
(iv) \(\frac{\partial \rho}{\partial t} + \mathbf{v} . \nabla \rho = 0\)

A.(i) and (ii)
B.(ii)
C.(i) and (iv)
D.(iii)

Exam Year(s)

GATE XE - 2025,

Answer: B

The mass divergence form is equation (ii) only.

Conservation of mass for a compressible fluid states that the local rate of change of density is balanced by the divergence of the mass flux:

\(\dfrac{\partial \rho}{\partial t} + \nabla \cdot (\rho\mathbf{v}) = 0\)

Equation (i) uses the curl instead of the divergence, and a curl is a vector while \(\partial\rho/\partial t\) is a scalar, so it cannot be a mass balance. Equation (iii) describes the time change of velocity, not of density. Equation (iv) says that density does not change following a fluid parcel, \(D\rho/Dt = 0\), which is the condition for incompressible flow; the full advective form of continuity is \(D\rho/Dt + \rho\nabla \cdot \mathbf{v} = 0\).

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In regions when there is a net outflow of air, a situation referred to as :

A.Convergence
B.Advection
C.Convection
D.Divergence

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: D

A net outflow of air is called Divergence.

Where more air leaves a region than enters it, the air is diverging. At the surface this happens under high pressure, where sinking air spreads out; in the upper troposphere divergence above a low draws air up from below.

Convergence is the opposite, a net inflow of air. Advection is horizontal transport of a property such as heat or moisture by the wind, and convection is vertical transfer of heat by rising and sinking air, so neither names a net outflow.

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Which quantity does horizontal divergence measure?

A.The rate of areal expansion of a fluid element
B.The rate at which a fluid element rotates about a vertical axis as it moves along with the flow
C.The circulation
D.The rate at which the shape of a fluid element is distorted while its total area remains unchanged

Answer: A

Divergence is the fractional rate of increase of the horizontal area of a fluid element, so positive values mean the element is spreading out and negative values mean it is contracting. Rotation about a vertical axis is vorticity, circulation is the loop integral of velocity, and area-preserving distortion is deformation. Together these four quantities describe every way in which the local velocity gradient can affect a fluid element, and separating them cleanly is the foundation of kinematic analysis in this subject.

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Which physical law connects horizontal divergence to vertical motion?

A.Conservation of potential vorticity, which holds whenever the motion is adiabatic and frictionless
B.Conservation of mass
C.Conservation of momentum, which requires that the forces acting on a parcel determine its acceleration
D.Conservation of energy

Answer: B

Mass continuity requires that air converging horizontally into a column must leave it vertically, so convergence at low levels implies ascent above. This is the single most used consequence of continuity in synoptic meteorology and it is why surface lows are cloudy and surface highs are clear. Momentum conservation is the equation of motion, energy conservation is the thermodynamic equation, and potential vorticity conservation is a derived result that follows from several of the governing equations combined.

Discuss in Forum

Which form does the continuity equation take in pressure coordinates?

A.The divergence of the mass flux vanishes at every point within the flow field being considered
B.The vertical integral of the divergence must equal the tendency of the pressure at the surface below
C.The horizontal divergence plus the vertical derivative of omega equals zero
D.The material rate of change of density is balanced by the divergence of the three dimensional velocity

Answer: C

In pressure coordinates continuity becomes a purely diagnostic relation with no density and no time derivative, stating that the three-dimensional divergence vanishes. This is one of the chief practical advantages of the coordinate, because it makes omega directly computable by integrating the horizontal divergence in the vertical. The other options give the height coordinate form, a general statement of mass conservation, and the surface pressure tendency equation respectively.

Discuss in Forum

Which quantity is omega in pressure coordinates?

A.The rate at which the height of a pressure surface changes at a point fixed above the ground below
B.The geometric vertical velocity
C.The rate at which the potential temperature of a parcel changes as it ascends through the layer
D.The material derivative of pressure

Answer: D

Omega is the rate of change of pressure following the parcel, so it is the pressure coordinate analogue of the vertical velocity. Because pressure decreases upward, omega is negative for ascent, a sign convention that catches many people out. The height tendency of a pressure surface is a local rather than a material derivative, and the rate of change of potential temperature is the diabatic heating, which is a thermodynamic quantity. Framing the question as which quantity the change acts upon makes the answer follow without calculation.

Discuss in Forum

Which sign does omega take when air is rising?

A.Negative
B.Positive, because the parcel is moving upward in the direction in which height increases from below
C.Zero
D.A sign that depends on whether the ascent is taking place above or below the level of the tropopause

Answer: A

Pressure decreases with height, so a rising parcel experiences falling pressure and omega is negative. Model output charts therefore show ascent, and hence cloud and precipitation, as negative values. The convention follows entirely from the coordinate being inverted relative to height, and remembering it as a consequence rather than as an isolated fact makes it much harder to get wrong when reading a chart in a hurry. The shared properties explain why these belong to one family, and the differences explain why more than one exists.

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Which relationship connects omega to the geometric vertical velocity \( w \)?

A.Omega equals w multiplied by the scale height of the atmosphere at the temperature of that layer
B.Omega is approximately minus rho g w
C.Omega is approximately equal to w divided by the density of the air at the level being considered
D.Omega equals g times w

Answer: B

Applying the hydrostatic relation to the material derivative of pressure gives \( \omega \approx -\rho g w \), where the small horizontal advection of pressure has been neglected. With a density near 1 kilogram per cubic metre near the surface, a vertical velocity of 1 centimetre per second corresponds to an omega of about 0.1 pascals per second. Keeping this conversion in mind allows model omega fields to be interpreted in physically familiar terms.

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10.

Which typical magnitude does synoptic scale omega have?

A.A few hectopascals per second
B.A few microPascals per second, which would be far too small to produce any cloud or precipitation
C.A few tenths of a pascal per second
D.A few pascals per second, which is comparable with the values found within a convective updraught

Answer: C

A vertical velocity of about 1 centimetre per second corresponds to an omega of roughly 0.1 pascals per second, so synoptic values are a few tenths of a pascal per second at most. Convective updraughts of several metres per second give values a hundred times larger. Knowing the characteristic magnitude allows an implausible model field to be spotted immediately, which is a practical skill worth having. Framing the question as which quantity the change acts upon makes the answer follow without calculation.

Discuss in Forum

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