Re: [Phys-L] When lightning strikes the ocean...

[John Denker <jsd@av8n.com>]

On 02/08/2018 05:06 PM, Derek McKenzie wrote:

> I am trying to understand (i.e. reconcile with established
> electromagnetic theory) what happens when lightning strikes the surface
> of a large body of water, such as an ocean. It appears that empirically
> the field/current dissipate mostly radially along the surface of the
> water, while only a small amount of energy makes its way vertically
> below the surface.

Interesting question.

> I have searched high and low for an authoritative model, but to no
> avail. Some invoke the Faraday Cage principle, whilst others invoke the
> Skin Effect, but nothing terribly compelling.

Executive summary:  I find the skin-depth argument to be
compelling.

In more detail:  Let me spell out the reasoning I used.

Step 1:  Use intuition to focus attention on avenues that
seem worth pursuing.  

Intuition 1a:  In 3rd grade I learned that current flows
in the entire bulk of the wire.  I remember the smiling
faces on the electrons in the diagram.

However, there is also intuition 1b:  I remember wiring
up an experiment for gigahertz frequencies at millikelvin
temperatures, using gold-plated insulators for wires,
rather than solid copper.  The reason is that the entire
bulk of the copper wire would conduct unwanted heat, but
only a thin surface layer carries the current at high
frequency.

So we have dueling intuitions.  The skin depth for lighting
hitting the ocean will be a lot longer, because the frequency
is lower.  Lightning is quick, but it is kHz quick, not GHz
quick.  Also the skin depth will be longer because the
conductivity of sea water is terrible compared to the
conductivity of metals.  On the other hand, the skin depth
depends only on the square root of the resistivity and
frequency, so maybe they don't matter quite as much.  Also,
the ocean is a lot bigger than a wire, so a depth that would
seem huge in a wire might seem small in the ocean.

Intuition 1c:  There was a time when the US Navy transmitted
ELF signals (76 Hz) to talk to submarines.  They wouldn't go
to that much trouble -- wavelengths of many thousands of km
-- if they didn't need have a problem with the skin depth of
sea water.

So we have no choice but to run the numbers.

Step 2:  A lightning bolt consists of multiple sub-strokes
each lasting on the order of 30 microseconds.  So the
characteristic frequency is tens of kHz, or about 200,000
radians per second.  The resistivity is on the order of
0.2 Ω⋅m.  Plugging into the skin-depth formula I get
something like 1.2 meters.  That's pretty thin compared
to the depth of the ocean.

You could argue that not all the energy is at that frequency,
and indeed there are frequency components all the way down
to DC, but even so, it's safe to say that most of the energy
dissipation will be in the top few meters ... for skin-depth
reasons alone.

========

The Faraday cage effect is real physics, but in this context
it seems like the answer to a different question.

The system can be modeled as a humongous spherical capacitor,
with the ocean as the inner plate and the ionosphere as the
outer plate.

At DC the charge will sit on the surface of the ocean, in a
layer with a thickness given by the Debye screening length,
which is on the order of nanometers for sea water.

However, that doesn't tell you anything about the route the
charge took to get there, or anything about dynamical effects
(including dissipation).