 We are going to have Professor Fukumoto first, but then the next speaker is Dr. Professor Ajetsu, as planned. So Professor Fukumoto is switched from tomorrow Friday to this session. And he's going to give us his talk here. OK. So the title of my talk is Gyroscopic Analogy of Coriose Effect for Stabilizing Rotating Stratified Floors Confined Analysphaloid. And we are considering about how our Rayleigh-Teller instability can be stabilized by rotation possible, try to get some insight into the mechanism. The effect of stabilization by rotation can be clearly understood by the dispersion relation of waves in a rotating stratified fluid. This is a continuous stratification with a brand visceral frequency is constant. Then you get following dispersion relations. This is Coriose Effect called inertial waves. This is gravity effect, this is internal gravity waves. Then without Coriose Effect, if n square is negative, it means unstable stratification, waves can be unstable. But by imposing rotations, system rotations, it tells you can stabilize the gravity waves. Then for short waves in 2D horizontal directions, gravity waves compete with winds through the rotation effect. But for long waves in the horizontal directions, stabilization by rotation can be possible. And we will see this effect in two ways. One is a toy model of a continuous stratification of linear flow confined in a spheroid. In the second part, we look into the Rayleigh-Teller instability. I was much influenced by this book by the late professor Dolce and Schiis. And the translator by both heathens describes how Poincare's idea is extended to the present day. Using a business approximations, you can construct some exact solutions of the Ibn Nabiya-Souk's equations. And consider a spheroid of the general spheroid. Then we assume incompressibilities. And density is divided into the uniform one and just small perturbations. And by taking the gradient of this continuity equation, you get a following form of the continuity equations for the gradient of the densities. And momentum equation or Euler equation in the business approximation can take this form by taking the color. And this looks like a torque and called Baroclingi torque. Then you can construct a steady solution of these business equations. This satisfies also the boundary conditions at the spheroidal walls. Then for general flows, you may pause following a solution form and substitute this into the Nabiya-Souk's equations because we assume linear flows. This question doesn't work. Then you obtain following the equations. It's constructed by multiplying these small omegas. A capital omega is actually the rotations. And also, actually, the vortices. Capital omega is vortices. And small omega is cooked from this. And this is the equation governing the heavy top in the frame fixed to the body. Omega is angular velocity. M is angular momentum. And this L0 is in the language of the body, just the axis from the fixed point to the gravity centers. And now, built from the frame fixed to the bodies, direction of gravity is rotating. This direction of gravity is represented by sigmas. In the language of fluids, sigma is coming from the gradient of the densities. And this top axis is associated with angle from the gravity direction of the principal axis. Then this torque comes from a Baruchnik effect. And this comes from the Coriolis effect. For example, for simplicity, if you consider upright spheroid, and then also consider the symmetric top. And when you apply the case, this top axis is simply directed in the g-axis, built from the body frames. Then there are two steady solutions. One is this component, this third component is one. This corresponds to the unsafe certifications. And this is actually sleeping top. And the other steady solution is this simply is gravity is just below the center of gravity is below the fixed point. Then as you may well know, stability of sleeping top is obtained in these following criterions by rotating faster than this critical speed. Sleeping top is stabilized. But if friction slower the rotation, and the rotation speed is smaller than this critical values, then tops just stop their rotating motion. Then motivated by this analogies, you can consider the stability of the flow confined in the tilted spheroid. And in this case, the top axis, it means the axis connecting from the fixed point to the center of gravity is misaligned with a symmetric axis. And this top axis built from the body coordinate looks like this. And this is actually when tilted, integrability is lost. And stability is influenced by this misalignment between symmetry axis and top axis. And you can easily find a steady state. This is just one parameter for families. And stability of this state can show how alignment effect phi non-zero will change the stability of the well-known sleeping top of Lagrangian case. First, we consider prolet case. Prolet case is elongated like a rugby ball. Then its feature is first inertia tensor, a component of inertia tensor is larger than a third one. And stability shows when there's no tilt, no misalignment, this is well-known critical point. And if you rotate top faster than this value, it is stable. When tilted, the stability region is larger than applied case. And more tilted, stability region is more tonically greater. And when just a tilting angle is half pi, entire region becomes stable. And this is a relation between the real part of frequency and the imaginary part. And this is a critical point. And this is a case of 60 degree of tilting angles. And when floating speed is slower than this value, where the two eigenvalues are degenerate, then instability starts. Again, another bifurcation occurs for very small values. Well, obliquely, there are more tricky behaviors. Actually, when denominator becomes 0, steady solution is lost. And first, if this is a gross rate, if a lot of speed is slower and slower, and if it becomes slower than these critical values, instability starts. And gross rate diverges. But at this point, steady solution is lost. But again, there is another bifurcation. And gross rate behave like this. And bifurcation also occurs around here. There is two regions. And by gaining insight into the stabilized mechanism by the analogy with top or gyroscopic effect, you can consider the two phase flows, the lateral instabilities. That's a simple upright sueloid. Just there is an idea. Rotation doesn't affect instability because this is unrelated with energies. But actually, this gyroscopic effect has a pronounced influence for the stabilizations. This is an interesting experiment made by Skiers of Nottingham. And rotation speed becomes 0 and becomes faster. Then by very ingenious method, they could realize the lateral environment. Then for faster rotation case, instability is delayed. And motivated by this, we consider the mechanism for the stabilizations. There are several pioneering works for rotational stabilizations. First, it was made by Carnibere and Professor Orlandy. And they focus on the effect of correlative force on vortex rings. Then there are recent works by Tower. And you see the Baldwin experiment. And also, they made beautiful theoretical works. Skip one slide. Then the experiment is a cylindrical apparatus. And we consider the two-layer fluid is confined in the lower half of the sueloid. This corresponds to considering the effect of topography. Heavy fluid is lying on top of light fluid. We consider oil equations in a rotating frame. And then there is equilibrium shape. And this interface shape is given by this form. And to consider this form is very important. And we consider stabilities. Then the equation for the small perturbation of the interface is given by Zeta. And you may consider following representations for the velocity using the potential. And then pressure is automatically obtained by this. And substituting these into incompressible conditions, you get a well-known equation governing potentials. This type of wave equation has inertial waves and inertial waves and internal gravity waves. The boundary condition is just flow through the wall. And also, no singularity on the central axis. And as the interface, pressure is continuous. And the kinematic boundary condition should be satisfied. And we want to know the growth rate. And then you may assume of the linear stability. Then you may assume the normal mode. Governing equation becomes this form. Then you may solve this equation with the boundary conditions in the following form. Yes, this is just for pressure continuity across the interface. Takes this form and kinematic boundary condition takes this form by substituting perturbation into the boundary conditions. And we use a variational method. It's initiated by miles. Then scarce and the collaborator uses this. And we follow these beautiful techniques. By multiplying the governing equation, this obtains a variation function, functionals. Actually, you can check by taking variation with respect to potential. You obtain the potential equation. Also, exact solution satisfies phi itself equals 0. Then by substituting a solution, partly satisfying this boundary condition on the walls. But without satisfying the boundary conditions in the interface, you may obtain by reducing this integral, you may obtain following form. Here, axiomatic mode is simple. And you may assume following form by taking n equals 0. And formally, this will constitute a complete set of functions. Then the parameter a is the radius around here is common, come from here. And you can substitute this and take variation with the amplitude c. And you obtain the equations. But for simplicity, you may first consider the very slow system rotation case. Then you can get dispersion relation for this slow rotation case. Here, a is the same as at number. Delta is aspect ratio of the swearloin. And this is gross rate with horizontal axis of rotation speed normalized in this. And this blue is the result of half swearloin case. And the red one is aspect ratio is 4.1. This is oblate swearloin. And horizontal axis is four times longer than just symmetric axis. And blue one is this one. And then more oblate. If oblateness increases, you have larger stable regions. And this dotted and this dashed is a cylindrical case derived by scales and their cloverators with same aspect ratios. Yes, delta. Delta is, I should write down delta. Delta is now one. Oh no, yes, I'm sorry. Delta is aspect ratio. This is 4.1 and 2.1. And this and this is same aspect ratio. And this and this aspect ratio is 2 and 1. Compared with swearloin and cylinders, you can see swearloin is more stable region has a wider than cylindrical case. For prolet swearloin, it means this symmetric axis is longer than horizontal axis. Then swearloin case and the cylindrical case is not much different. Red one is 1.4, blue one is 1.1. This is the interpretations for elongated or prolet and more elongated means inertia moment in the horizontal direction is greater than inertia moment in the actual directions. Then critical speed for stability of sleeping top is increased. Therefore, more elongated is more unstable. And oppositely, if flattened, this is more overlaidness is increased. It means if she's increased, top is more stable. And you can interpret this from the stability of this top. Then finally, you can calculate the critical rotation rate for swearloin to establish stability of these two layer fluids. Then critical speed means this omega gross rate imaginary part omega equals 0. Then you may assume following. Then by taking this in power series of this small parameter and substitute this approximation, approximated eigenfunctions into this and taking the variations, you obtain critical rotation speed. Then the result is following. If you assume the same parameter used by experiment of scales, then you obtain critical rotation speed for the swearloin and this aspect ratio, I should write the aspect ratio somewhere, is this parameter. And I think almost one, 0.7 something, but I should write down. Then with same aspect ratios, you obtain the following critical rotation speed. And the case, the critical rotation speed for making establish stability is following. Then swearloin case can establish stabilization with lower rotation speeds. This is just the same graph as we showed previously. Summarize, we consider how a courier's effect can stabilize or can sustain heavy fluid on top of a light fluid. In the first part, we consider uniformly certified flow or linearly certified flow with velocity proportion linearly in coordinates. Then we looked at stability of the flow in tilted swearloin. And this is the same as the stability of the heavy symmetric top with top axis and the symmetric axis misaligned. Then in the second part, we consider regular instability of rodent flow confined in an upright half swearloin compared with a finite cylindrical case. Swearloin has a more stable environment. Thank you very much.