Hurricane size vs planet rotation speed

Coriolis' force is the cause for winds and water streams to bend while they travel North-South along the Earth surface. This force is inversely proportional to the rotation speed of the planet: the quicker the planet rotation, the higher the force.

Now, we know that in the past the day was shorter, because the rotation speed was higher. Assuming all other parameters to be the same, how would this reflect on an hurricane? Let's say during the Ordovician, when the day was 21 hours long.

Due to the higher Coriolis' force, would the hurricane be more compact or not? Would this then concentrate the energy on a smaller area resulting in a stronger hurricane?

• I don't think a higher Coriolis force would make bigger hurricanes. In fact, Earth's rotation rate being faster, 21 hours, not 24, there might be less day-night temperature variation which might mean smaller temperature variation. The atmosphere might have been denser during the Ordovician period too, and the oceans warmer and, larger when there was one continent and one big ocean. A very fast planet rotation might create a more permanent hurricane like formation but, in and of itself, 21 hours vs 24, I don't think rotation would be much of a factor. Hard to say with certainty though. – userLTK Mar 15 '17 at 23:25

The Coriolis Effect's contribution to wind speed is negligible

The acceleration due to the Coriolis force is $$\mathbf{a} = -2\mathbf{\Omega} \times \mathbf{v},$$ where $\mathbf{\Omega}$ is a vector with magnitude equal to the angular velocity of the coordinate system ($7.29\times10^{-5} \text { rad/s}$), and diretion equal to the axis of rotation--that is from the Earth's south pole to north pole. $\mathbf{v}$ is the velocity of the air in question.

The Coriolis effect is a fictitious force, that is, it is not a real force acting on an object but it is an artifact of calculating motion in a non-inertial frame of reference. In order to estimate the effects of the coriolis force on a hurricane, we can relate this Coriolis force to another fictitious force, the centrifugal. The centrifugal force is the important (fictitious) force in keeping objects in orbit by balancing out the pull of gravity. We can set up an analogy for how a hurricane works by setting the Coriolis force equal to the centrifugal force for a theoretical air particle rotating around the edge of the hurricane, to see how fast an air particle will rotate if its motion is dominated by the Coriolis effect.

The centrifugal force is $$\mathbf{\omega} \times (\mathbf{\omega} \times \mathbf{r}),$$ where $\omega$ is the angular velocity of the hurricane (note the omegas in the equation should be bold vectors, that is not showing up in formatting), and $r$ is the distance from the center of the hurricane. We can drop the vectorization by assuming particle motion is in a circle on a plane with axis of rotation of the hurricane perpendicular to that plane.

The velocity of a rotating object is $$\mathbf{v} = (\mathbf{\omega} \times \mathbf{r})$$ which is looking familiar from our centrifugal force equation. For the Coriolis force, I argue that the result of the cross-product, in order to be perpendicular to both $\mathbf{\Omega}$ and $\mathbf{v}$, is in the same direction as the centrifugal force vector. Therefore we cancel the $(\mathbf{\omega} \times \mathbf{r})$ term, leaving only the $\sin{\theta}$ for the angle between the Earth's axis of rotation, and a plane on the planet's surface. $\theta$ is the latitude; the factor is 0 where the angle is 0 at the equator and thus there is no Coriolis effect, and 1 at the poles. We solve for the magnitude of the $$|\mathbf{w}| = |2\mathbf{\Omega}\sin{\theta}| = 0.73\times10^{-4} \text{ rad/s}$$ for a latitude of 30 N.

What does this number mean? If we multiply that rotational speed by a distance from the core, we should find the component of expected wind speeds caused by the Coriolis force. Thus if we multiply by 100 km, we get expected winds speeds of 0.73 m/s.

Conclusions

First off, we can point out that if we use 1000km, we get expected wind speeds of 7.3 m/s, which is pretty significant. However, this is only a rude approximation, hurricanes have to follow the rules of fluid dynamics and such as well. The eye of a hurricane extends for many kilometers to either side, so this approximation is not useful at all at smaller distances, and hurricanes don't have enough energy to drive winds that fast out to 1000km (at least not on Earth). In real life, the maximum wind speed is sustained at the eyewall, at the boundary of the eye. The radius of maximum wind is around 50 kilometers.

At distances of 50km and up, our above equation gives Coriolis-caused wind speeds of less than 1 m/s. Obviously, a hurricane is much more powerful than this. A category 1 hurricane on the Saffir-Simpson scale has sustained wind speeds of 33-42 m/s.

Overall, the ability of the Coriolis effect to hold together a hurricane is small, around 2 orders of magnitude lower than the main contributer, which is the pressure gradient caused by rising hot air. From this Wikipedia link you will see that the Coriolis effect is a pre-requisite for cyclone formation. There needs to be sufficient force to develop the smooth, spinning, fluid flow in the atmosphere; a turbulent flow to the low pressure point will prevent cyclone formation.So, while the Coriolis effect is critical to the formation of a hurricane, it is not nearly powerful enough to cause 35 m/s + winds to spiral into a compact cyclone.

Given this, we can assume that we would need roughly two orders of magnitude increase in Earth's rotational velocity before the Coriolis effect can significantly affect the size and wind speed of a hurricane.

• I like your answer, and, correct me if I'm wrong, but the wind speed by the Coriolis effect increases with size, so if you have a great red spot like "hurricane", 1/10th the diameter of the planet or over 1,000 miles across, you'd need less than 2 orders of magnitude. But otherwise, I agree with your answer. The engine that drives hurricanes is warm oceans, and warm moist air rising, creating a low pressure system. The Coriolis effect helps hurricanes get their start and determines their direction of spin but it's oceans, temperature and air pressure that is the engine. – userLTK Mar 15 '17 at 23:19
• @userLTK Yes, math error there. I fixed that, thank you. – kingledion Mar 15 '17 at 23:43
• My idea was not that the Coriolis' force would increase wind speed, but rather "curl more" the wind streams and therefore resulting in the vortex being more wrapped. – L.Dutch - Reinstate Monica Mar 16 '17 at 8:11
• @L.Dutch Tracking. My argument is that the inwards 'curl' caused by the Coriolis force is insignificant compared to the inward 'curl' caused by the low pressure cell. I demonstrate this only inferentially, by showing that the coriolis force can only 'curl' a wind of 0.7 m/s at 100 km distance from the eye. Since there are winds more like 50 m/s + that are being 'curled,' the pressure gradient must be much more significant in shaping the hurricane. Therefore the Coriolis force will have to increase by orders of magnitude to affect hurricane shapes. – kingledion Mar 17 '17 at 0:48
• @kingdelion how about you bring some of this math here worldbuilding.stackexchange.com/questions/123419/… and make this planet windy using Coriolis effect? – Willk Sep 6 '18 at 12:12