Wallace Luchuk, President
Penurious Engineering Inc.
First Posted 9-6-98
Note: This paper has minor revisions in preparation. Part of these revisions include correcting the Greek mathematical symbols commonly used in the aerodynamic equations.
The vorticity field of a tornado is examined from an aerodynamicist's point of view. Sources of the vorticity are considered and evaluated. A new source for the driving vorticity is proposed. It is suggested that the energy of the tornado comes from the inter- action of a storm generated toroidal electric current field with the earth's magnetic field (a J x B force). The intensification of the tornado when touching down is explained by the stimulating effect of the earth's conductivity. Evidence supporting the theory is presented, and methods of enervating the tornado are suggested.
Tornadoes are a very well known caprice of our meteorological environment. Considerable effort has been expended to discover and to alert the communities of their presence. This is a necessary and commendable effort! However our understanding of the physics of the tornado in its full blown form is painfully inadequate and there is no current defense against them. There are a number of good references (Snow 1984, and Davies-Jones 1995) that describe the meteorology leading up to the development of the tornado, but candidly admit to a void in explanations for the fully developed tornado. This paper examines the physics of the developing and full blown tornado from an aerodynamicist's point of view as to what possible forces are available to drive this incompressible meteorological phenomenon. A new theory is presented that proposes that the tornado is driven by the interaction of an organized electric current field with the earth's magnetic field. The driving force, a J x B body force, is intensified when the tornado touches down by the higher conductivity of the earth.
The tornado, in essence, is a very strong vortex with one end attached to the ground and the other end extending upward to a varying or undefined height. This configuration suggests to the aerodynamicist that there are vortex lines concentrated in the core of the tornado, that are splayed out radially in all directions on the ground. The vortex lines are diffused upward into the clouds only to turn downward and connect up with the vortex lines on the ground at some remote distance from the tornado. The whole vortex field appears to be toroidal in shape with one end flattened by the ground. This picture is based on the aerodynamic theory that vortex lines must close on themselves and cannot be unconnected.
The next point of consideration is: Where is the source of these vortex lines?
In aerodynamic theory, vortices are usually generated by the shear in a fluid with a velocity gradient, such as in a boundary layer. The generation of strong vortices require strong velocity gradients, and most strong vorticities are usually created by the interaction between an established fluid flow and a solid surface. On a wing, for example, there is an array of vortex lines of one polarity generated by the boundary layer on the upper surface, and another array of opposite polarity generated by the boundary layer on the lower surface. The vorticity on the upper surface is much stronger than the lower surface because of the higher velocity and resulting higher shear forces in the boundary layer. At the wingtip, the two vortex arrays merge and the vortex that leaves the wingtip is the sum of the two sets of vortices. In any event, the energy that is dissipated in the boundary layer and contributes to the strength of the tip vortices come from the free stream. The flow between the solid surface and the fluid has been driven by some external power source. In flight, the engines drive the aircraft through the air providing the relative motion between the wing surface and the air. In the wind tunnel, the wind tunnel compressor provides the power of the airstream that flows past a stationary model.
Somewhere along the closed loop of vortex lines of the tornado there is an energy source that gives rise to the vorticity of the tornado. If the energy source is purely meteorological, then there must be strong velocity gradients somewhere in the tornado meteorological field that generates the vorticity of the tornado. This is in contrast to normal aerodynamic principles in that there cannot be strong velocity gradients in free air air, because there is no mechanism to support them.
For the strong vorticity at the ground to be a consequence of the meteorological field, two items must be present: 1. Air must have a transport property that is capable of delivering the immense power of the tornado to the ground: and 2. There must be a transport mechanism or configuration that uses this transport property and effects the transfer. Because of the dilute properties of the atmosphere, neither of the above items are available!
Another point of consideration is that whenever there is an energy system, the energy density always decreases with distance from the source. This is decreed by the second law of thermodynamics that says that energy cannot be transmitted from a source without accompanying losses. Since the most energetic location of the rotational kinetic energy of the tornado is at its base, this is the location of the source of this energy. Not in the clouds!
The analysis of the tornado in its steady state form requires that it be considered in a state of equilibrium. This means that any and every element of the tornado has, on, or within the element, both driving forces and retarding forces, that are in balance. If we examine the forces on a pie shaped element of the tornado, as shown in Figure 1, there is a centrifugal force due to rotation that must be balanced by a pressure gradient, and there must be equal and opposite driving and retarding tangential forces.
Since we do not fully understand what the driving force is we must look at the retarding force. As aerodynamicists, if we know the velocity profile, we can calculate the shear or retarding force, - both, if the tornado is completely in the air, and if it is on the ground. In the air, the shear force is very weak because the air viscosity is very small and the velocity gradients are also very small (compared, for example with an aircraft boundary layer).
Calculations show that the dissipative power for a tornado in air with a simple velocity profile (increasing linearly in the core and decaying hyperbolically outside the core), a maximum velocity of 90 meters/sec (295 ft/sec) and a radius of 30.5 meters (100 ft)and a height of 305 meters (1000 ft) is of the order of 1 horsepower (See Appendix 1). However, calculations of the drag horsepower of a turbulent boundary layer of the same velocity profile (as described above) when the tornado is on the ground, yields tens of thousands of horsepower (See Appendix 2). One need not be reminded of the incredible strength that tornadoes can have on the ground. Since there is little energy in the air column portion of the tornado and intense energy on the ground, this again suggests that the source of this rotational energy is close to the ground.
If as stated above, the tornado is in equilibrium, each or any element of the tornado that we choose to examine is in equilibrium. This tacitly implies that where the retarding forces are weak the driving forces are also weak, and when the retarding forces are strong, the driving forces must also be strong! - This is the crux of the analysis!
This means that the strong driving forces are developed immediately at the air-ground interface where the retarding shear forces are the greatest! This is where the strong vorticity of the tornado is generated!
Now the problem becomes: "What can be the driving force at the air- ground interface?"
There is one force that heretofore has not been considered as a driving source for the tornado, and that is a J x B body force. This is the same force that drives electric motors and would imply that the tornado is a "natural" electric motor. This force is composed of an electric current density, J, and a magnetic field, B (See Appendix3). When these two are at right angles to each other, the J x B force is developed at right angles to both of the components.
The next question is: "Where does this J x B force come from?"
One element of the tornado that has been considered in the past is the considerable electrical activity associated with the storm that generates the tornado (Vonnegut 1960, Whiting 1964, and Davies- Jones 1975). However, most of the studies have been concerned with the ohmic heating of the air due to the electric current and the effects of the magnetic field of the electric current flow itself. The results of these analyses proved negative.
It is proposed here that the driving source of the tornado is a toroidal electric current field (or a partially toroidal field) that may be generated by the apparently random atmospherics of some storms, and its interaction with the earth's magnetic field. The toroidal electric current in this static electric field is very similar to the vorticity toroidal field. If the core of the tornado is considered to be the core of the electric current toroid, the reduced pressure in the core would tend to increase the electric current density (J) due to the increase in air conductivity. Air conductivity increases as the air pressure falls. The inflow (or outflow) of current, in air, to the base of the tornado core is almost at right angles to the earth's magnetic field and the interaction of the two provides a tangential J x B force.
One difference between the toroidal vorticity field and the toroidal current field is that at the ground-air interface, the vorticity lines cannot penetrate the earth, while the electric current lines can. One other difference that must be mentioned here is that the subsonic aerodynamic field adjusts itself to meteorological perturbations at the speed of sound, while the electric field adjusts itself to electrical field perturbations at close to the speed of light.
Usually, the severe electrical part of a storm has passed before the tornado appears, and there is little evidence of lightning in the general area surrounding the tornado. (Evidence is available in Davies-Jones, 1975, of lightning coming out the bottom of a tornado, and the tornado often glows suggesting electrical activity within the tornado.) This suggests that the part of the storm with the severe voltage gradients has past, and the voltage gradients in the area surrounding the tornado and in the toroidal electric current field are not so severe as to induce lightning strikes. Around most tornadoes there are few lightning strikes in the vicinity of the tornado. Recent satellite data has revealed that lightning in the vicinity of a tornado mysteriously ceases when the tornado is being spawned.
If the toroidal current stays in the air, a nascent tornado may appear, because it does not require much power to sustain a tornado in air. However when the tornado touches down, the tornado becomes very powerful. The intensification of the tornado at touchdown can be explained by the difference between the air conductivity and the ground conductivity. Air conductivity is about 6 x 10-14 mhos/meter (Nat. Acad. Press. 1986), while the ground conductivity is about 10 x 10-3 mhos/meter (Weeks, 1964). This very large difference in conductivities causes the intensification of the electric current (J is the current density) at the ground-air interface. This is the location where the J x B driving force (in air) would be the greatest, and the location of the greatest retarding force. The ground current flowing into the air follows a curving upward path into the core, and any radial component of this current that is not aligned with the earth's magnetic field will generate a tangential body force within the air. This is proposed to be the energy source that drives the fully developed tornado. Upper and lower limits of this force may be estimated from estimates of the limits of the current density. The current density can be obtained from:
J = k * dE/dl Where: J = Current density (amps/m2) k = Conductivity (mhos/m) dE/dl = Voltage Gradient (volts/m)
The voltage gradient under a storm prior to a lightning strike is a maximum ofabout 30 kilovolts per centimeter (the breakdown voltage of air - see Cobine, Reference) and using the air conductivity and the air voltage gradient yields a current density (Ja) of 1.8 x10^-7 amps per square meter. However if we use the earth conductivity and the air voltage gradient, the current density (Je) becomes 3 x 10^4 amps per square meter. This is an extremely excessive current density! The true ground voltage gradient must be considerably less than the air voltage gradient. Upper and lower limits of the J x B force may be made using Ampere's Law that states:
F = BIL/10 (Ref: Haussman & Slack)
and in vector form:
F = B x IL/10
F = Force on a conductor (dynes) B = Magnetic flux density (gauss) I = Conductor current (amps) L = Conductor length (centimeters)
In larger metric units:
F' = 100B x IL'
F' = Force on a conductor (Newtons) L' = Length of conductor (meters)
This conductor force can be converted to a body force by multiplying by A/A:
A = Conductor cross-sectional area (meters2) F' = 100B x (I/A)L'A = 100 B x JL'A and F'/V = 100B x J
F'/V = Body force (Newtons/meter3) J = Current density (amps/meter2)
In English units this equation becomes:
Fe/Ve = 1459 B x Je
Fe = Force in pounds Ve = Volume in cubic feet Je = Amps per square foot
This final equation permits the estimation of the body force in air.
To estimate the upper and lower limits of the J x B force we can use the earths magnetic field strength (from .14 to .28 Gauss in the U.S.) and the above current densities. Using B (.28) and Ja we get 2.52 x 10-9 Newtons/m3 as a force per unit volume. Using B and Je we get 420 Newtons/m3 as the body force. The first value is way too small, and the second value is way in excess of that required to drive the tornado. Any body force approximating the air density (.126 Newton/m3) would be more than sufficient to drive the tornado. The true current at the air-ground interface must be somewhere between the two.
Another way to estimate the maximum force necessary to drive the tornado is to estimate what the maximum retarding force in the tornado. Since the only retarding force, in air, is the viscous force, the maximum viscous body force can be estimated from the maximum shearing stress: Tau(max) = Mu * (dVr/dr)(max)
Where in English units:
Tau(max) = the maximum shearing stress (lbs/ft^2) Mu = Air Viscosity (lb.sec./ft^2) Vr = Rotational Velocity (ft/sec) r = Radius (ft) dVr/dr(max) = Maximum velocity gradient
The maximum velocity gradient in a tornado with a simple profile of a linear increase in veocity with radius in the core and a hyperbolic decrease in velocity with radius outside the core (see Appendices)is at the edge of the core and is equal to Vr/r. Air viscosity using Sutherland's equation for air at 80 degrees F is equal to 3.856*10^-7 lb.sec./ft^2. If we assume the tornado has a maximum velocity of 460 ft/sec ( the maximum recently recorded velocity) and a core radius of 100 ft, the maximum velocity gradient is 4.6 ft/sec.ft. From these values we find that the maximum shear stress is: Tau(max) = 1.776*10^-6 (lbs/ft^2) If it is assumed that this stress is in effect over one foot of radius, then a retarding force per unit volume, F', similar to a body force is obtained. This retarding body force is opposed by the J x B driving force and permits the computation of the current density, J (amps/ft^2). Using the equations in English units of Appendix 3: F' = F/V = 1459.4 * (J x B) Since the earth's magnetic field varies from .14 to .28 Gauss, the current density J, is: J = (1.776*10^-6)/(1459.4*.28) = 4.329*10^-9 (amps/ft^2) Which is a very small current density required to overcome the maximum air resistance! To determine if this is a realistic value of current density, we will try to determine if any other physical variables of problem are violated. Using this current density and the air conductivity we can compute the air voltage gradient from: E/l = J/ka Where: E = Voltage (Volts) l = Conductor length (ft) ka = air conductivity (6.58*10^-14 mhos/ft) Then: E/l = 4.329*10^-9/6.58*10^-14 = .660*10^5 (Volts/ft) or 66,000 volts/ft This value is well below the air breakdown voltage of 914,000 Volts/ft (30,000 Volts/cm, - Ref. Cobine). Therefore it appears to be possible to drive the air in the tornado to its maximum velocity without incurring a voltage breakdown or lightning strike!
The addition of energy by J x B forces (that are directed kinetic energy forces) must increase the total enthalpy and total pressure of the air. However most of this energy is added in some kind of activity layer at the air-ground interface and varies with radius and height in some fashion. Future experiments to try to measure tornado variables should include total temperature, and total pressure measurements immediately at ground level, as well as the ground currents or other electric field sensors such as magnetometers. The current practice of measuring wind velocities, in and about the tornado using dopler radar, is measuring the effect, and reveals nothing about the cause!
If the proposed theory is true, it should be possible to detect and determine the tornado strength by measuring ground currents over an area surrounding the tornado. The ground current around a tornado should be in a radial or a partially radial pattern. The center of this radial pattern should locate the tornado, and the direction of the current (inward or outward) should reveal the direction of rotation of the tornado.
One other implication of the theory is that if the earths' magnetic field is a requirement, then there should be no tornadoes (or waterspouts)along the earth's magnetic equator, because the earth's magnetic field (being a dipole) would be horizontal there. Also if the tornado strength is a function of magnetic field strength and current density, there should be a correlation between areas with the strongest magnetic field strengths and ground conductivities, and the strongest tornadoes.
If the above described physics of the tornado is correct, then suitable defensive measures may be devised. Reducing, shorting or controlling the electric field (either in the air or in the ground) or so that the ground current would be returned to the air alligned with the magnetic field vector, or reducing the local magnetic field would enervate the tornado. Depriving the tornado of its energy source would cause it to come to an end shortly on the ground because of the considerable ground drag, and the tornado would appear to rise up into the air where the air dissipation is so much less.
In the absence of any other relatively complete explanation for the intensification of tornadoes when touching down to the ground, the proposed theory fits many of the observations and provides a more complete model of where the primary source of energy that drives the fully developed tornado comes from. This model might direct future students of tornadoes to direct their measurement efforts toward confirmation (or refutation) of the theory. Once the theory is confirmed, realistic defensive hardware might be devised.
Davies-Jones, R.P. and Golden, J.H., "On The Relation of Electrical Activity to Tornadoes", Journal of Geophysical Research, Vol.80, No.12, 1975.
Davies-Jones, R.P., "Tornadoes", Scientific American, Vol.273, No.2, August, 1995.
Snow, J.T., "The Tornado", Scientific American, Vol.250, No.4, April, 1984.
Vonnegut, B., "The Electrical Theory of Tornadoes", Journal of Geophysical Research, Vol.65, pp203-212, 1960
Weeks, W.L., "Electromagnetic Theory for Engineering Applications", p282, John Wiley & Sons, New York, 1964.
Wilkins, E.M., "The Role of Electrical Phenomena Associated With Tornadoes", Journal of Geophysical Research, Vol.65, No.12, 1964.
National Academy Press, "The Earth's Electrical Environment", Washington, DC, 1986, Fig.12.15, p176.
Hausmann, E., and Slack, E.P., "Physics", D. Van Nostrand Co. Inc, New York, 1943
White, F.M., "Fluid Mechanics", McGraw-Hill Publishing Co., New York, 1986, Eq. 7.45, p402
Cobine, J.D., "Gaseous Conductors, Theory and Engineering Applica- tions", Dover Publications,Inc., New York, 1958, Fig.7.7, p164.
To evaluate the retarding or dissipative power of the air about a tornado, a simple radial variation of the tangential velocity is assumed. This profile has a linear velocity variation in the core (up to the maximum velocity), then a hyperbolic velocity decay outside the core. The equations for this profile are below.
CORE EXTERIOR Vr = Vo(r/ro) Vr = Vo(ro/r) Where: Vo = The maximum tangential velocity (m/sec.) ro = The radius at maximum Velocity (meters) Vr = Velocity at radius, r (m/sec.) r = Local radius (meters)
The local dissipative or drag force on a cylindrical area is the shear force due to air viscosity and is defined by:
Dr = Tau*A = Mu*(dVr/dr)*(2*Pi*r*h) Where: Dr = Drag force on the cylindrical area (Newtons) Tau = Shear stress in air (Newtons/m2) Mu = air viscosity (Newton sec./m2) h = tornado column height (meters) Pi = 3.14159
If the drag force is multiplied by the local velocity, then a local dissipative power may be obtained;
Pr = Dr*V/405.65 Where: Pr = the local dissipative power (hp) 1/405.65 = conversion constant (Newton meters/sec to HP)
Since the power varies with radius, we must first differentiate with respect to r:
dPr/dr = (1/405.65)[Dr(dV/dr) + V(dDr/dr)]
Integrating the local power from r = 0 to r = ro and from ro to infinity in two separate integrals, and noting that the negative value obtained for the second integral denotes a direction only and not a negative dissipation, the following result is obtained:
P = 4*Pi*Mu*h*Vo^2/405.65
Now putting in some physical values, an estimate of the dissipative power in air of the tornado may be obtained:
Using: Mu = 1.8203 x 10-5 (Newton sec./m2) r = 60.96 (meters) (not relevant) Vo = 91.44 (m/sec.) h = 304.8 (meters) P = 1.436 horsepower
As in Appendix 1, if the velocity profile of the tornado on the ground is known, then the ground drag or ground dissipative power may be estimated. The equation for the drag power over an annulus of ground area is:
Pr = Vr*Dr/405.65
Where the variables are the same as before. However the turbulent skin-friction ground drag is defined in aerodynamic form as:
Dr = Cd*q*A = Cd*(rho*Vr^2/2)*A Where: Dr = Drag force on the local annulus (Newtons) Cd = Turbulent drag coefficient (non-dim.) q = Local dynamic pressure (Newtons/m2) rho = Air mass density (Newton sec.2/m4) Vr = Local averaged tangential velocity (m/sec.) = [Vr(n+1) + Vr(n)]/2 A = Local annular area (m2) = Pi*(r(n+1)^2 - r(n)^2)/4
Now the turbulent skin-friction drag coefficient may be defined as:
Cd = 0.031/(Rer)1/7 (Ref: White) where: Rer = Local Reynold's number based on the radius (non- dim.) = Vr*r*rho/Mu
and the other variables are as defined in Appendix 1. Now the local radius is not the appropriate dimension to use in determining the Reynold's number, but since the Reynold's number grows so fast with radius, the drag coefficient is not strongly affected.
The local power equation (see Appendix 1) can be numerically integrated to obtain the total dissipative power:
P = Sum((dPr/dr)*dr)
This has been done and the results are shown Table 1 and are plotted in Figure 2.
The force on an electrical conductor is: Fe = BIL/10 (Ref: "Physics" by E Hausmann & E.P Slack, Van Nostrand Co. 1939) Where: Fe = Force on conductor (Dynes) B = Magnetic field strength (Gauss) I = Conductor current (Amps) L = Conductor length (Centimeters) Since: 1 Newton = 100,000 Dynes 1 meter = 100 Centimeters Fe' = BIL'*(100) Where: Fe' = Force on conductor (Newtons) L' = Conductor length (meters) Rearranging and deviding by the conductor cross-sectional area Fe'/(L'*A') = Fe'/Ve' = B*(I/A')*100 = B*J'* 100 Where: A' = Conductor cross-sectional area (meters^2) J' = Current density (amps/meter^2) Ve'= Conductor volume (meters^3)
Now converting this equation into english units; 1 Newton * .2248 = 1 pound 1 meter = 3.28088 ft 1 meter^2 = 10.7639 ft^2 1 meter^3 = 35.3147 ft^3 So our above equation becomes: Fe''/Ve'' = Fe'*(.2248/Ve')*35.3147 = B*J*(100/10.7639)' Fe''/Ve'' = 1459.45* B* Je' where: Fe'' = Force on conductor (lbs) Ve'' = Conductor volume (ft^3) Je' = Current density (amps/ft^2)
Note:This paper was last revised on June 7, 1999. W.L.