Symmetry Breaking in Fluid Motion

First uploaded on 2022/10/06
Last updated on 2023/04/25
Copyright(C)2022 jos <jos@kaleidoscheme.com> All rights reserved.

When you start working on field physics, you will first study the basic mathematics used as a tool. This is so-called vector analysis, and divergence and convergence are one of the first concepts to appear there. As an example, a schematic diagram of water or air blowing out from a point in space filled with the same medium is shown, which is said to be divergence. Similarly, a schematic diagram of water/air being sucked in from a point in space is shown, which is said to be convergence. The flow pattern remains the same for both outflows and inhalations. If time could be reversed, the flow pattern would remain the same, only the direction of flow would be reversed. This is easy to see and intuitively understand, and in fact, it provides a correct picture for electric and magnetic fields, for example: reversing the arrangement of the electric charges or magnetic poles only reverses the direction of the electric or magnetic field, but the field pattern remains exactly the same. However, when the medium that flows out or is sucked in is actual water/air, this diagram is highly misleading.

Textbooks showing such examples are always accompanied by a note that says, "when water/air is blown out or sucked in quasi-statically". The term "quasi-static" is very difficult to understand, but it may refer to a situation in which "water/air seeps out or seeps in without violent motion". When water/air is blown out or sucked in with very normal motion, the situation is not as simple as the textbooks make it out to be.

Attach a nozzle to a point in the space and let water/air blow out from it. For very weak flows, it may be possible to reproduce the phenomena as in the textbooks. However, beyond a certain velocity, the flow transitions to turbulence, which cannot be handled by simple mathematics. In the case of a blowout, the larger the flow velocity, the faster and more complex the transition to turbulence becomes. When the flow becomes turbulent, momentum and mass transport efficiency drop dramatically. This is the reason why water accidents rarely occur in swimming pools near water supply outlets.

On the other hand, let us attach a nozzle to a point in the space and inhale water/air from there. In this case, there is no turbulence in the space. No matter how large the suction flow rate (with a velocity below the sound speed) is, the flow remains laminar throughout the space. A laminar flow can transport momentum and mass much more efficiently than a turbulent flow. As a sad example, major accidents occur every year near the drains of swimming pools. This is due to the incredibly strong forces generated by the laminar flow in the vicinity of the drain. Even the best pool designers should recognize the difference between an inflow (which may involve turbulence) and a discharge (which invariably involves laminar flow). Once again, pool managers should inspect for hazards near the drainage outlets.

Thus, reversing the time of a flow field in which a blowout has occurred is not the same as a flow field in which only a suckout has occurred, and they can be easily distinguished.

The reason for this difference in treatment between electromagnetic and fluid fields, even in the same field physics, is due to the difference in the characteristics of the fields, i.e., whether they are linear or nonlinear. In the case of linear fields, such as electromagnetic fields, it is possible to discuss them using simple conventional vector mathematics and assuming symmetry. They are detailed in textbooks and are used in examinations. On the other hand, nonlinear fields are still being studied. Typical examples of nonlinear fields are the motions of water/air. It is surprising that the motions of water/air, which are most familiar to us, remain as a final area that we do not yet understand. They are only briefly mentioned in textbooks. We are trained primarily with linear fields and have limited knowledge of nonlinear fields. It should be especially noted that there is no guarantee that the symmetries observed in a linear field will also be observed in a nonlinear field.

As mentioned here and there on this website, dynamical detrainment is currently being studied as a mechanism of cumulus outflow. This concept was first proposed around 1970. Research on nonlinear phenomena had already been conducted at that time. However, only the specialists, i.e., researchers in nonlinear mathematics and nonlinear physics, were strongly aware that their research targets were nonlinear phenomena. Although nonlinearities are inherent in most phenomena occurring in nature, it may have been a rare case that researchers other than the specialists were particularly conscious of a nonlinearities latent in their research subjects. Moreover, the most of the research subjects for nonlinearity at that time were nonlinear waves. Wave phenomena always have a restoring force (e.g., dispersion in a soliton and dissipation in a shock wave) that balances the nonlinearity, so to some extent, conventional treatments of wave phenomena could be applied to them. However, when the nonlinearity becomes stronger and the flow becomes messy (with no restoring force any more), conventional methods can no longer be used. Convection, which is the motion of cumulus clouds, is a typical example of strongly nonlinear motion without restoring force.

Because cumulus clouds are familiar phenomena in our daily lives, we tend to think of them as something that can be easily handled. However, we need to be keenly aware that cumulus clouds are one of the most strongly nonlinear phenomena occurring in the atmosphere. We shouldn't expect a simple symmetries in it. For example, the words "dynamical detrainment" sound to us just like a play on words, the opposite of (dynamical) entrainment. It is natural that (dynamical) entrainment occurs when a thermal plume mixes with the surrounding air. Because mixing is nothing other than (dynamical) entrainment. However, to think that there is the opposite, dynamical detrainment (blowout), during or after mixing, is too much of a failure to take into account the symmetry breaking of fluid motion. As long as the atmosphere is a nonlinear medium, it is quite possible that only entrainment occurs and no dynamical detrainment at all. This is because it is impossible for fine vortices filling a cumulus domain to integrate themselves into a flow on a large physical scale, a phenomenon that goes against the second law of thermodynamics.

RDC (Radiatively Driven Circulation) we have proposed is a mechanism that assumes that a large atmospheric field extending outside the cumulus determines the flow from the cumulus. This makes thermodynamical sense in that a larger physical field determines a smaller flow field inside. Furthermore, note that RDC is not a blowout from a cumulus cloud, but a suction from a cumulus cloud due to the vacuum that forms in the surrounding atmosphere. If the actual air is blown out from the cumulus clouds, turbulence will be created, which will reduce the efficiency of air transport. But if air is sucked out to everywhere in the atmosphere, the flow stays laminar throughout the entire atmosphere, and air can be transported very effectively out of the cumulus domain.