### The Structure Of Symmetrical Depressions.

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*Introduction and Summary*.

Before the advent of Frontal Analysis, prognosis of the movement of pressure systems on the weather chart, insofar as it was based on other than empirical considerations, relied on theories of the mutual interaction of identifiable (radially symmetrical) vortices. This approach yielded little of value for practical forecasting, partly because it ignored the role of asymmetric systems, but also, partly, because unsatisfactory models of the symmetrical depression itself were used as a basis for the theory.

Frontal theory, correctly, stresses the importance of discontinuity in the short-term prognosis, and its very success in this restricted field has drawn attention away from the necessity of coming to grips with symmetrical systems, which still, in large measure, govern the general long-term weather situation. The time seems ripe, therefore, for a reappiaisal of the role of the symmetrical vortex in general meteorological theory.

The classical picture^{1} envisages a depression as a central mass of air in solid rotation, surrounded by a region in which wind speed varies inversely as the distance from the centre. Such a structure can be made to fit actual depressions reasonably well over a limited range, but it must be excluded as a valid overall picture of vortex structure on theoretical grounds. The wind speed falls off so slowly with radial distance that a vortex of infinite extent would possess an infinite kinetic energy.

It is possible to produce a simple model of wind and pressure structure tree from these faults.

The model investigated predicts that the ratio h/V_{m} R_{m} for a cyclone should only vary within narrow limits. Here V_{m} is the maximum wind-speed at a distance R_{m} from the centre, and h is the depth of the depression.

An examination of 18 depressions (Table 1) shows that this ratio does, indeed, vary within narrow limits.

The total kinetic energy for a vortex can be evaluated, and also the mutual kinetic energy of adjacent vortices.

The mutual interaction of vortices presents features of great interest. Two vortices of like sign attract for large separations, but repel for close approach, while vortices of unlike sign repel and then attract. Thus we have a phenomenon analogous to the barrier effect of wave-mechanics. Several features of the interaction of cyclones, anticyclones, and fronts, on the synoptic chart can be interpreted in terms of this barrier effect.

The investigation is mainly confined to geostrophic vortices, i.e., vortices in which the pressure gradient is balanced by the coriolis force due to the earth's rotation. A brief discussion is also given of the cyclostrophic vortex, in which the pressure gradient is balanced by the centrifugal acceleration of rotating winds. This model applies to the tropical cyclone, and accounts satisfactorily for the well-known “eye” phenomenon. A similar structure might also be postulated by smaller-scale vortical systems, such as tornados, and the small-scale vortices of turbulence phenomena.

*The Specification of a Vortex.*—Meteorologists customarily speak of depressions as deep or shallow according as to whether the central pressure is very low, or not so low, in relation to the average pressure over a wide area. A depression may also be categorised as extensive, or widespread, or small. These are qualitative categorizations. The meteorologist is not obliged to put figures to depth or extension, and, indeed, may not have a perfectly clear idea of precisely what is meant by these terms. Nevertheless, his intuitions are right. The salient properties of a symmetrical vortex, considered merely as a distribution of pressure about a point, are something corresponding to a depth, and something corresponding to an extension, just as the distribution of any statistical quantity may be characterised in part by specifying a mean value, and some parameter representing the spread.

For refined analysis these two parameters may not be sufficient, but for any consideration at all they are necessary. It is necessary, therefore, to inquire

[Footnote] 1. See Rayleigh, *P.R.S.* (A), No. 93, p. 148; Brunt, *P.R.S.* (A), No. 99, p. 397.

what quantities, obtainable from the synoptic chart, may be taken as measures of depth or spread of a depression. The central pressure might be taken as a measure of depth, but this is not going quite far enough; what we want is depth in relation to some observed data. We might, perhaps, define the depth as the difference of the central pressure and the average pressure for the area for that particular season. This gets closer to the mark, but it ignores the fact that, even apart from the existence of a particular depression, the pressure over a wide area may be abnormally high or low in relation to the statistical average. The conclusion is that it is not possible to measure depth directly from the synoptic chart. It is possible, however, to measure a parameter which depends on the depth.

In an atmospheric vortex the pressure-gradient is zero at the centre and at large distance therefrom. Somewhere in between it reaches a maximum value, depending both on the depth and the spread of the vortex. The radius of maximum pressure-gradient. which coincides, under geostrophic balance, with the radius of maximum wind-speed, may itself be taken as measure of the extent of the depression. There are thus two parameters of a cyclone, the radius of maximum wind-speed, R_{m}, and a maximum wind-speed, V_{m}, from which it is possible to infer the strength and spread of a vortex, given the form of its pressure-profile.

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The pressure profile of a radially symmetric vortex must be such that the pressure-gradient is zero at the centre, and falls away at large distances sufficiently rapidly for the total kinetic energy of the system to be finite. A pressure profile satisfying these conditions is
p = p_{0} + he^{−r2/a2} (1)

where p_{0} is the pressure at great distance from the centre, r = 0. At r = 0, p = p_{0} + h. so that h (negative) may be defined as the depth of the depression. The parameter h will be positive for an anticyclone. The parameter a is a measure of the spread of the vortex. As e^{−r2/a2} decreases very rapidly with r, when r is a multiple of a, the influence of the vortex is completely lost at a relatively short distance from the centre. The wind-speed in this type of vortex drops off rather more rapidly with distance than is usual in depressions, but the model has the merit of mathematical simplicity.

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An alternative type of profile is represented by
p = p_{0} + h/1 + 1^{n}/a^{n} (2)

where, in addition to the two parameters h and a, we have a third parameter n, the index of the distribution. Here wind-speed falls off less rapidly with 1 than is the case with a normal vortex.

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*The Normal Vortex*: The tangential wind-speed in a geostrophic vortex at distance r from the centre is:
V = ½ρω òp/òr (3)

where ρ is the density of air and ω the vertical component of the earth's rotation, the sense of the motion being clockwise in the southern hemisphere, anti-clockwise in the northern for a vortex of low pressure. Substituting the value of òp/òr obtained from the equation (1) in (3) we have
V = h i e^{−12/a2}/ρωa^{2} (4)

The radius of maximum wind-speed is given by òv/òr = 0, i.e., R_{m} = a/√z. The radius of maximum wind-speed is thus 0.707 times the spread, a. The maximum wind-speed obtained by making the substitution r = R_{m} in (3) is V_{m} = h e^{−½} / 2ρ ω R_{m}

We infer from this that the ratio
h/V_{m}R_{m} = 2 √e. ω ρ = 3.30 ωρ (5)

is invariant for all “normal” geostrophic vortices in that latitude.

The kinetic energy of unit volume of air is
½ ρ V^{2} = h^{2}r^{2}e^{−2r2/a2}/2ρω^{2}a^{4}

The total energy of a horizontal disc of air, thickness dz is
π h^{2}dz/ρω ∫^{∞}_{0} x^{2} e^{−2x2}dx = π h^{2}dz/8 ρω^{2} (6)

Integrating in the vertical, we find for the total kinetic energy of the vortex
π/8 ω^{2} ∫^{∞}_{0} h^{2}dz/ρ (7)

If we assume that the depth of the depression varies with height according to the equation h = h_{0} (1-z/Z) for z < Z, and h = 0 for z > Z, we find for the total kinetic energy of the vortex system
π h_{0}^{2} Z/16 ρ ω^{2} (7a)

where ρ is some mean density of the air column up to height Z.

We see from (7) that the total kinetic energy of a normal vortex varies as the square of the depth, the height in the atmosphere to which the vortex extends, and inversely as the square of the vertical component of the earth's rotation.

One remarkable feature of this result is that the kinetic energy of a normal geostrophic vortex does not depend at all on the spread of the disturbance.

We infer from (7) that the kinetic energy of a vortex increases on movement towards the equator, and decreases on movement towards the pole, provided the depth and vertical extent of the depression remain the same.

One might anticipate latitudinal movement of depressions to be associated with compensating changes in h_{0}, or Z, or both. Thus, if a depression were to move from latitude 30° to latitude 45°, the energy of the system would remain unchanged, if the depth in latitude 45° were 1.4 of the depth in lat. 30°, with Z unaltered, or if the vertical extent were double with h_{0} unchanged.

*The Power Vortex*: Similar results may be derived for a vortex with the pressure profile given by equation (2).

Here we have for the tangential wind-speed,
V = n h/2 ρ ω a^{n} r^{n−1}/(1 + r^{n}/a^{n})^{2} (8)

The radius of maximum wind-speed is given by
[R_{m}/a]^{n} = n − 1/n + 1 (9)

For the maximum wind-speed we have
V_{m} = (n^{2} − 1) h/8n ρ ω R_{M} (10)

Thus we have
h/V_{M} R_{M} = 8n/n^{2} − 1 ρ ω (10a)

For n = 2, the characteristic ratio of the vortex, h/V_{m} R_{m}, is equal to 5.3, for n = 3, 3.0, and for n = 4, 2.1. It becomes small for large values of n.

Table I gives the values of V_{m}, R_{m} and h_{0} for 18 depressions, as estimated by Goldie.^{2} Goldie's values of V_{m} have been increased by 1.4 to allow for the effects of surface friction. The fourth column shows the ratio of V_{m} R_{m}/h_{0} and the last column, the value for n, inferred from this ratio in the expression (10a), correct to a half-integer.

The experimental data are subject to large errors, but one can infer from the clustering of values of n around 2 or 3 that a reasonable fit to extra-tropical cyclones can be obtained with a pressure profile of the type (2). The data do not permit us to say whether n is the same for all vortices, or varies from vortex to vortex.

The small spread of the characteristic ratio encourages the belief that some such structure as that envisaged holds for extra-tropical cyclones. The “classical” theory permits h/V_{m} R_{m} to have any value at all.

^{2}

[Footnote] 2. Goldie, *Geophys. Mens.*, No. 79 (1939).

The total kinetic energy of a vortex is given by
2π ∫^{∞}_{0} x^{2}dz ∫^{∞}_{0} x^{2} ½ ρ V^{2} r dr

Taking the value of V^{2} from (8) we find for the total energy
nπ/24 ω^{2}∫^{∞}_{0} x^{2} h^{2}dz/ρ = nπ h_{0}^{2}Z/48 ω^{2} ρ (11)

using the same assumptions regarding the vertical extent of the depression as in the last section.

The expression (11) for the energy of a power vortex is exactly analogous to the corresponding expression (7) for a normal vortex. except that the new parameter n enters. The energy of a power vortex for n = 3 is, indeed, exactly the same as for a normal vortex, and the characteristic ratios are nearly the same. For many purposes a normal vortex can be substituted for a power vortex, index 3.

*The Interaction of Vortices*: If we have two adjacent vortices giving kinetic energy fields
½ ρ V_{1}^{2} and ½ ρ V_{2}^{2}

the resultant kinetic energy field is,
½ ρ (V_{1} + V_{2})^{2} = ½ ρ V_{1}^{2} + ρ V_{1}.V_{2} + ½ ρ V_{2}^{2} (12)

The total kinetic energy of the system is thus the sum of the kinetic self-energies of the individual vortices, plus the cross-term ρ V_{1}.V_{2} representing the mutual kinetic energy of the interacting vortices.

It is our purpose, now, to consider this mutual term.

We consider only the normal vortex, on the grounds of mathematical expedience, as this type of vortex gives a solution in simple terms. The results obtained will remain valid, in a qualitative sense, for vortices of the power type.

Consider two normal vortices of the same strength, h, and spread, a, at points (O, O) and (x_{0}, O) respectively.

We may write for the cross term
ρ V_{1}.V_{2} = h^{2} r^{2}/ρ ω^{2} a^{4} e^{−[r2+′2/a2]}−h^{2} × x_{0}/ρ ω^{2} a^{4} e^{−[r2+′2/a2]} (13)

where r^{2} = x^{2}, + y^{2}, r′^{2} = (x − x_{0})^{2} + y^{2}.

Integrating this expression over all space, and assuming h = h_{0}(1−z/Z), for z < Z, as before, we have for the mutual energy of the vortices
E = π h_{0}^{2}Z/8 ω^{2}ρ e^{−x02/2a2}[1−x_{0}^{2}/2a^{2}] (14)

We notice that for two vortices of the same sign E is negative for x_{0} > √2a, and positive for x_{0} < √2a. On approaching, two vortices of like sign lose from their joint stock of kinetic energy, up to a distance √2a, which energy reappears in the form of isallobaric kinetic energy associated with the approach of the vortices. Vortices of like sign therefore attract for separations larger than √2a, and repel for closer approaches.

Vortices of unlike sign, on the other hand, are mutually repelled for large separations, and attracted for close approaches.

This interesting “barrier” effect, which inhibits the over-close approach of two vortices, permits new interpretations of a number of synoptic situations. For example, a series of eastward-moving frontal depressions is frequently terminated by an outburst of polar air; the cold front pushes far to the north, with an anticyclone building up in the cold air in its rear, while the previously existing anticyclone in the warm air ahead of it collapses simultaneously.

This situation can be interpreted in terms of the mutual interaction of the cold front, considered as a line vortex, with the warm anticyclone. If the line vortex approaches the circular vortex with sufficient speed, it can crash through the potential barrier, temporarily annihilating the anticyclone by producing a counter-circulation, and pass out of the potential barrier on the northward side of the high, the circulation of which is re-established in the cold air. On this view, there is no real annihilation of the anticyclone with a corresponding creation of a cold high. The one anticyclonic system exists all the time, and is only temporarily eclipsed.

*The Tropical Cyclone*: So far we have been concerned exclusively with geostrophic vortices. In low latitudes geostrophic control is too weak to balance a strong-gradient, and in tropical cyclones the balance is provided, in the main, by the centrifugal force of rotating winds.

The equation for balance between centrifugal force and the pressure gradient is
ρ*V*^{2}/r = -òp/òr (15)

If we assume the pressure-profile (2) the equation (15) becomes
ρ*V*^{2} = n h r^{n}/a^{n}/(1 + 1^{n}/a^{n})^{2} (15a)

The total kinetic energy of such a vortex is,
n π a^{2} ∫^{∞}_{0}h dz ∫^{∞}_{0} r^{n+1} dr/a^{n+2 (1+rn/an)2} (16)

For n = 4 this expression becomes
π^{2} a^{2}/2 ∫^{∞}_{0}h dz (16a)

The kinetic energy of a centrifugal vortex is proportional to the depth, to the vertical extent, and to the square of the spread.

The integral (16) diverges for n < 2, so that the index for a tropical cyclone must be greater than 2.

The maximum velocity, at 1 = a, is
V_{m} = ½ √ n h/ρ (17)

From (17) we infer that
V_{m2}/n h = ¼ρ

is an invariant for centrifugal vortices.

In Table II is listed values of V_{m}, h, and the values inferred for n for 18 tropical cyclones. It will be remarked that the dispersion is considerably greater than for the approximately geostrophic vortices of temperate latitudes.

The values of n are also higher on average. This is, perhaps, what might be anticipated from the well-known “eye-of-the-storm” often associated with tropical cyclones. The fact that a calm can exist over a wide area, with adjacent hurricane winds, indicates that the rate of wind increase around the centre is less than would be associated with a solid rotation.

Since wind-speed is not a linear function of the pressure-gradient in centrifugal vortices, it is not possible to develop a theory of the mutual energy of adjacent vortices along the lines of the interaction of vortices. The theory of centrifugal vortices is thus of necessity more limited.

Cyclone No. | h (mb) | V_{m} (m/sec.) |
R_{m} (km) |
h/V_{m}R_{m} ρ ω |
n |
---|---|---|---|---|---|

1 | 43 | 27.5 | 550 | 4.0 | 2.5 |

2 | 48 | 27.5 | 500 | 5.0 | 2 |

3 | 43 | 27.5 | 450 | 5.0 | 2 |

4 | 52 | 40.5 | 280 | 6.4 | 2 |

5 | 55 | 35.2 | 420 | 5.25 | 2 |

6 | 61 | 37.3 | 450 | 5.0 | 2 |

7 | 21 | 25.2 | 360 | 3.2 | 3 |

8 | 9 | 26.6 | 320 | 1.5 | 5 |

9 | 21 | 28.0 | 380 | 2.8 | 3 |

10 | 33 | 25.5 | 470 | 3.8 | 2.5 |

11 | 25 | 18.2 | 480 | 4.0 | 2.5 |

12 | 45 | 36.5 | 320 | 5.5 | 2 |

13 | 55 | 37.0 | 430 | 4.8 | 2 |

14 | 41 | 28.2 | 460 | 4.5 | 2.5 |

15 | 63 | 37.0 | 500 | 4.8 | 2 |

16 | 68 | 33.6 | 650 | 4.4 | 2.5 |

17 | 48 | 32.4 | 700 | 2.9 | 3 |

18 | 47 | 33.6 | 400 | 5.0 | 2 |

No. | Place | Date | (h) (mb) |
V_{m2} (m^{2}/sec^{2}) |
V_{m2}/h_{o}) (gm^{−1}cm^{3}) |
n |
---|---|---|---|---|---|---|

1 | Dry Tortugas | 9/9/19 | 77 | 2900 | 380 | 2 |

2 | Miami | 18/9/26 | 73 | 7920 | 1080 | 5.5 |

3 | Bermuda | 22/18/26 | 47 | 6720 | 1420 | 7.5 |

4 | Key Largo | 28/9/29 | 60 | 9220 | 1540 | 8 |

5 | East Columbia | 13/8/32 | 67 | 4240 | 640 | 3.5 |

6 | Savanna | 5/11/32 | 93 | 16900 | 1820 | 9.5 |

7 | Cape Henry | 23/8/33 | 40 | 1920 | 480 | 2.5 |

8 | Turks | 1/9/33 | 30 | 3920 | 1240 | 6.5 |

9 | Jupiter | 3/8/33 | 60 | 6480 | 1080 | 5.5 |

10 | Cape Hatteras | 16/9/35 | 51 | 2380 | 800 | 2.5 |

11 | Bimini | 23/9/35 | 63 | 6040 | 960 | 5 |

12 | Miami | 1/11/35 | 37 | 2320 | 620 | 3 |

13 | Fort Walton | 31/7/36 | 37 | 3360 | 920 | 5 |

14 | Manila | 20/10/1882 | 40 | 6040 | 1000 | 8 |

15 | Manila | 13/9/28 | 76 | 10600 | 1400 | 7.5 |

16 | Japan | 21/9/34 | 100 | 7200 | 720 | 3.5 |

17 | Indo China | 16/7/28 | 50 | 3720 | 740 | 4 |

18 | Hong Kong | 18/8/23 | 50 | 2740 | 550 | 3 |