Simple Models of Atmospheric Vortices

Atmospheric circulation systems have a wide variety of structures and there is no single mechanistic model that describes all their characteristics. However, we can construct simple kinematic models that capture some primary aspects of the flow. For simplicity, we will concentrate on idealized extra-tropical depressions. We will not consider hurricanes and tropical storms in any detail, because the effects of moisture condensation and convection dominate their behaviour.

Stationary Flow

For steady flow ({\partial/\partial t = 0}) of an axisymmetric circular cyclonic vortex, the balance of forces is given approximately by the gradient wind relationship:

\displaystyle \underbrace{\frac{V^2}{r}}_{ \mathrm{CFG} } \quad+ \quad \underbrace{fV}_{ \mathrm{COR} } \quad - \quad \underbrace{{\frac{1}{\rho}}{\frac{\partial p}{\partial r}}}_{\mathrm{PGF}} \quad = \quad 0 \,. \ \ \ \ \ (1)

Here {V=V(r)}, with {r} the distance from the centre, is the speed of circulation around the centre {r=0}, {f=2\Omega\sin\phi} is the Coriolis parameter, with {\Omega} the earth’s angular velocity and {\phi} the latitude, {\rho} is the density and {p} the pressure. We will take {f} and {\rho} to be constant and assume that the radial component of velocity vanishes. The first term in (1) is the centrifugal force (CFG), the second is the Coriolis force (COR) and the third is the pressure gradient force (PGF).

Balance of Forces

Geostrophic Balance

The ratio of the first two terms in (1) is the Rossby number:

\displaystyle \mathrm{Ro} = \left[ \frac{\mathrm{CFG}}{\mathrm{COR}} \right] = {\frac{V}{fr}} \,.

In many cases this is small, and we will assume this and ignore the centrifugal term. What remains is called the geostrophic wind:

\displaystyle V = V_{\mathrm{GEOS}} = {\frac{1}{f\rho} }{\frac{\partial p}{\partial r}} \,. \ \ \ \ \ (2)

Insofar as this holds, the wind field is determined completely by the pressure field and vice versa. If we know the pressure pattern, we can easily determine the wind, which blows along the isobars, lines of constant pressure. In the Northern Hemisphere {f>0}, and the low pressure is to the left if you stand with your back to the wind: the flow is anticlockwise around a low pressure centre.

Cyclostrophic Balance

Another interesting limiting case is when the Coriolis force term is small. Then the Rossby number is large, and there is approximate balance between the centrifugal and pressure gradient terms terms:

\displaystyle \frac{V^2}{r} \approx \frac{1}{\rho} \frac{\partial p}{\partial r} \,. \ \ \ \ \ (3)

This is called cyclostrophic balance. Since {V} occurs as a square, the flow may be clockwise or anticlockwise.

Inertial Balance

The third balance relationship, rare in the atmosphere but common in the ocean, occurs when the pressure gradient force is small. Then there is balance between the centrifugal and Coriolis terms:

\displaystyle \frac{V^2}{r} \approx- f V \qquad\mbox{or}\qquad \frac{V}{r} \approx f \,. \ \ \ \ \ (4)

For circular motion, the frequency is {\omega = V/r} so we have {\omega \approx f} for this inertial balance.

Special Vortices

Case I:

For typical atmospheric circulations, the radial pressure gradient is reasonably close to a constant. Thus, let us assume the pressure field is

\displaystyle p = p_0 + \gamma r \,,

where {\gamma} is constant. The isobaric surface is an inverted cone with apex at the circulation centre. The geostrophic wind speed is also constant:

\displaystyle V = \frac{\gamma }{ f \rho} \,.

This is the simplest assumption we can make, and is a reasonable first approximation to what is found in nature. However, there is a singularity at {r=0}: it is clear that the horizontal gradient of the wind blows up there. Moreover, {V} is independent of {r} and {\mathrm{Ro}\rightarrow\infty} as {r\rightarrow 0}.

The vertical component of vorticity in cylindrical polar coordinates is

\displaystyle \zeta = \frac{1}{r}\frac{\partial (rV)}{\partial r} =\frac{\partial V}{\partial r} + \frac{V}{r}

In the present case, this yields

\displaystyle \zeta = \left(\frac{\gamma }{ f \rho}\right)\frac{1}{r} \,.

Variation of pressure with distance r from the vortex centre for the four cases discussed in the text.

Case II:

To avoid the singularity, we might assume a paraboloidal isobaric surface:

\displaystyle p = p_0 + \frac{1}{2}\gamma r^2 \,,

where {\gamma} is constant. Then the geostrophic wind is given by:

\displaystyle V = \left(\frac{\gamma }{ f \rho }\right) r \,.

The wind vanishes at {r=0} and increases in proportion to {r}. The angular velocity of the flow is {\omega = V/r = \gamma/f\rho}, constant. Thus, we have solid body rotation, that is, the fluid rotates as if it were a rigid body. The vorticity is constant:

\displaystyle \zeta = \left(\frac{2\gamma }{ f \rho}\right) \,.

There is no singularity, but the pressure pattern is unrealistic: we do not find that the pressure gradient of cyclones continues to increase as we move out from the centre.

Case III:

We can get the best of both worlds by assuming a pressure field of the form

\displaystyle p = p_0 + \gamma [ \sqrt{1+r^2} -1 ] \,,

where again {\gamma} is constant. The isobaric surface looks like an inverted cone with a rounded apex. Close to the centre, {r} is small and the pressure is given approximately by

\displaystyle p = p_0 + \frac{1}{2}\gamma r^2

as in Case II. The circulation is approximately solid body rotation. Far from the centre, the pressure is given approximately by

\displaystyle p = p_0 + \gamma [ r - 1 ] \,,

the radial pressure gradient is constant as in Case I and the speed of rotation around the centre is approximately independent of radial distance. The angular velocity {\omega = V/r} decreases for large {r}.

Variation of windspeed with distance r from the vortex centre for the four cases discussed in the text.

Case IV:

Finally, we consider a piecewise differentiable pressure field:

\displaystyle p = p_0 + \frac{1}{2}\gamma r^2 \, , \qquad r < a \,;

\displaystyle p = p_0 + \frac{1}{2}\gamma a^2 + \gamma a^2 \log \left({\frac{r}{a}}\right) \, , \qquad r > a \,.

This comprises an inner and outer region. The geostrophic wind is given by:

\displaystyle V_{\rm INNER} = \left({ \frac{\gamma}{ f\rho} }\right) r \, , \qquad V_{\rm OUTER} = \left({ \frac{\gamma a^2}{ f\rho} }\right){ \frac{1}{r} } \, .

Clearly, the maximum {V=\gamma a/f\rho} is at {r=a}. The inner motion is solid rotation, like Case II. The vorticity is constant {\zeta=2\gamma/f\rho} in this region. In the outer region, the wind speed gradually decreases to zero. The vorticity vanishes identically in this region.

{\star\quad\star\quad\star}

Case III is a reasonable approximation to reality for extra-tropical depressions. However, we must remember that actual atmospheric circulation systems are generally not axisymmetric. In the extra-tropics, the structure of cyclonic vortices is strongly modified by frontal systems, approximate discontinuities in the temperature and wind fields while, in the tropics, water plays a dominant role, complicating matters significantly.

Moreover, in tropical regions the Coriolis term is small, so that the approximations made above are invalid. However, Case IV is at least a crude first approximation to the structure of a tropical storm, with solid rotation within the eye, maximum wind at the eye-wall and wind speed gradually decreasing further out.

For a simple model of a frontal depression, see an earlier post here.

Sources

{\bullet} Doswell, Charles A., 1984: A kinematic analysis of frontogenesis associated with a nondivergent vortex. J. Atmos. Sci., 41, 1242–1248.

{\bullet} Holton, James R., 2012: An Introduction to Dynamic Meteorology, Academic Press; 5th Revised edition, 552 pages. ISBN: 978-0-1238-4866-6.

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