[chris (OP)]

I was asked the other day by someone where do gusts of wind come from? He was referring specifically to the sudden increases in wind intensity that appear out of nowhere, rather than the steady blowing of the window.

[evan_au]

On the large scale (100 km or more), winds are driven by temperature differences, pressure differences, moisture differences and the Coriolis force from Earth's rotation. This produces the High and Low pressure regions that we see on weather maps.
See: https://en.wikipedia.org/wiki/Weather#Causes

You can get one-time changes in wind strength, direction and temperature when a cold front or warm front passes through. This is associated with the moving boundary between the large-scale structures mentioned above.
See: https://en.wikipedia.org/wiki/Cold_front#Development_of_cold_fronts

On small scales (<1km), these large scale winds interact with hills, trees and buildings on the ground to produce turbulent flow, which produces semi-random eddies and pauses, down to very small scales.
See: https://en.wikipedia.org/wiki/Turbulence

For aviators like Alan, storm cells can also produce ordered flows of air, which produce sudden changes in wind speed and direction when you fly through it at high speed in an aeroplane. This is called turbulence by pilots.

[RD]

I think the gusts large eddies, (bigger than the house), where laminar flow has become turbulent ...
https://en.wikipedia.org/wiki/Eddy_(fluid_dynamics)

[chris (OP)]

Is it not also a consideration that if you get a bunch of air masses arriving at the same point, at the same time, then the wind that one feels at that point will be the vectorial sum of the movements of all of the air masses. So could it be that a gust occurs when air masses meet and one of their directional components becomes particularly large?

[evan_au]

the vectorial sum of the movements of all of the air masses

The velocity and pressure of a gas (eg air) can be calculated using the Navier-Stokes equations.

These are a set of nonlinear differential equations which are difficult to solve, even on supercomputers. One of the great unsolved questions in maths & physics relates to solutions to this complex set of equations.

But scientists persist with these equations (and approximate solutions to them) for important applications like predicting the weather, ensuring aeroplanes will fly, improving jet engines, designing buildings and minimising petrol consumption of cars (or maximising speed without becoming a glider, in the case of Formula 1 racing).

Because they are nonlinear, you can't model air movements as a simple vector sum - especially when factors like condensation and evaporation of moisture plays such an important part in clouds and storms.

Solutions to these equations are chaotic and unpredictable (in the mathematical sense) - even the smallest deviation in the initial conditions will, over time produce very different outcomes - the legendary Butterfly Effect.

See: https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations#Properties

[Colin2B]

At a local level gusts depend on the stability of the air in the region between ground and the cloud base. Stable air is associated with strata (layer) clouds and unstable with cumulus (fluffy) clouds.
Cumulus clouds which often form following a cold front passing are due to warm, moist rising air. As it rises the air cools and will fall, when it hits the ground it spreads outwards and as Chris says will add vectorially to the surface wind so depending on direction of each will either increase or reduce the surface wind - these will be felt as either a gust or lull.
You will often find small cumulus clouds forming rows - called cloud streets - in the direction of the upper wind. Air rises under the clouds in the centre of these streets and falls in the areas of clear sky - circulating back towards the centre of each cloud street - so you can usually make a reasonably accurate prediction of the pattern of gust/lull as you sail through these areas.
With larger cumulus clouds you would tend to get strong gusts where air falls from the leading edge (in wind direction) and during a thunderstorm these can be pretty violent as the air can fall from a considerable (colder) height.
All of this is much easier to observe in a flat area e.g. at sea, as Evan says, on land hills, buildings trees etc change the patterns making them less predictable. Also over land surface friction slows the air reducing the gusts and also changes the direction of the air as it falls.