M&P2 的最后一讲的Lecture的 3.Equation of motion for fluid 部分的完整转录稿

 完整转录稿如下：（我tm不懂啊

Shallow Water Modelling – Part 2

Now, let's think about the equation of motion for a fluid. If you recall the advection equation, this is a PDE or partial differential equation that describes the Lagrangian derivative, which is d/dt. In an Eulerian framework by breaking it down using the chain rule, as partial df/dt, plus the velocity multiplied by partial df/dx. And this is a one-dimensional advection equation. And we use this in previous weeks to look at the transport of rain between levels in a 1d column model, for instance.

As we've been talking about, Newton's second law states, F equals ma or ma is equal to F. And we can write that as m times dV by dt because the acceleration is just the rate of change of velocity with respect to time. Now, here we have a Lagrangian derivative dV by dt, and for our model describing the motion of a fluid, we want to use an Eulerian framework. So we have to change this Lagrangian derivative into an Eulerian derivative. Our model is going to be a two-dimensional model.

So not only do we have the rate of change of F with respect to time and the rate of change of F with respect to x, we also need a rate of change of F with respect to y, the second dimension. And so when we apply this advection equation to V, what we get is something that looks like this. Remember, we have two equations because this is a vector equation. So we have an equation of motion for the x direction, 3.11, and then equation of motion for the y direction, which is this 3.12.

Now, let's look at the left-hand side first. We've got this term rho, which is the density of the fluid. Now that's analogous to the mass in our F equals ma for a fluid. And that density is multiplied by this acceleration term here. It's like an advection equation for the velocity. Now, I've broken down the velocity into a velocity along the x direction, and the velocity along the y direction. The velocity along the x direction is called u and the velocity along the y direction is called v. So we have this acceleration term here, which is just the advection equation in an Eulerian framework.

On the right-hand side, you have the forces acting on the fluid. So we have a term here, dP by dx. That's a pressure gradient term, P is the pressure; it's a change in pressure over x. And in the second equation, we have dP by dy, changing pressure of y. And we also have two extra terms here and here, and these are forces that contain the velocity along the y direction and the velocity along the x direction. This is known as the Coriolis effect as we'll see later.

The last terms on the right-hand side of equations 3.11 and 3.12, minus dP by dx, minus dP by dy, are pressure gradient forces, PGF. They are negative because a negative pressure gradient gives rise to a positive force. So think about our window: the forces from high pressure to low pressure, not from low pressure to high pressure. So these are negative when they're describing forces. And they can be estimated using the equation for PGF, which is the change in pressure, delta P, divided by a distance, either delta x or delta y. So alone, the pressure gradient force – what does it do? Well, let's look at figure 3.1. If you've got a region of high pressure shown by this H and a region of low pressure in our fluid. And these numbers here refer to the pressure. So we've got high, 1030, decreasing with distance away from the H. And we've got low, 1002, increasing with distance away from the L. So this is low pressure. This is high pressure. The pressure gradient force alone acts from high to low pressure. As shown, by this red arrow here. So taken alone, what would happen is if we have some fluid at this position here, it would be directed towards the L. You just move in a straight line towards the L.

Now, the first term on the right, in equations 3.11 and 3.12, this rho fv and this minus rho fu describe the Coriolis force or Coriolis effect. Now, let me explain this by looking at this animation on the left-hand side here. So let's imagine we're looking down on Earth from space. Which is the upper part of this picture here. And let's imagine there's a black ball which is at the North Pole and there's an observer, which is this red dot here, positioned some distance away from the North Pole. There's somebody stood at the North Pole and they throw this black ball away from the centre, then it will move in a straight line as observed from space, and that's because it obeys Newton's laws of motion and just move in a straight line. Because there are no forces acting on the ball once it leaves the person's hand. But to an observer stood some distance away from the North Pole rotating anti-clockwise with the Earth, they would observe that the ball actually bends towards the right of its motion. It would follow a curved path.

浅水建模 - 第2部分

现在，让我们考虑流体的运动方程。如果你记得平流方程，这是一个偏微分方程（PDE），描述了拉格朗日导数，即d/dt。在一个欧拉框架中，通过使用链式法则，我们可以将其分解为partial df/dt，加上速度乘以partial df/dx。这是一维平流方程。在先前的几周中，我们使用这个方程来研究例如一维柱模型中雨水在不同层次之间的输运。正如我们所讨论的，牛顿的第二定律表明F等于ma或ma等于F。我们可以将其写为m乘以dV/dt，因为加速度只是速度相对于时间的变化率。现在，在这里我们有一个拉格朗日导数dV/dt，而对于描述流体运动的模型，我们要使用欧拉框架。因此，我们必须将这个拉格朗日导数转换为欧拉导数。我们的模型将是一个二维模型。因此，我们不仅需要关于时间和x方向的F的变化率，还需要关于y方向的F的变化率，即第二个维度。因此，当我们将这个平流方程应用于V时，得到的结果类似于这样。请记住，我们有两个方程，因为这是一个矢量方程。所以我们有x方向的运动方程3.11，和y方向的运动方程3.12。现在，让我们首先看左边。我们有密度rho这个术语，它是流体的密度。现在，这类似于流体的F等于ma中的质量。这个密度乘以加速度项，类似于速度的平流方程。现在，我将速度分解为沿x方向的速度和沿y方向的速度。沿x方向的速度称为u，沿y方向的速度称为v。因此，我们在这里有这个加速度项，它只是欧拉框架中的平流方程。在右边，你有作用在流体上的力。因此，我们有一个术语dP/dx。这是一个压力梯度术语，P是压力，它是x方向上的压力变化。在第二个方程中，我们有dP/dy，即y方向的压力变化。我们还有两个额外的项在这里和这里，这些是包含沿y方向和沿x方向的速度的力。这被称为科里奥利效应，我们稍后会看到。方程3.11和3.12右侧的最后两项，减去dP/dx和减去dP/dy，是压力梯度力（PGF）。它们是负数，因为负的压力梯度产生正向的力。所以想象一下我们的窗户：从高压到低压的力，而不是从低压到高压。所以这些在描述力时是负的。它们可以使用PGF的方程来估算，即压力变化ΔP除以一个距离，要么是Δx要么是Δy。因此，压力梯度力——它是做什么的呢？好吧，让我们看看图3.1。如果你有一个显示高压的区域，如图中的H，和我们流体中的低压区域。这些数字指的是压力。所以我们有高压，1030，随着离H越远而减小。还有低压，1002，随着离L越远而增加。这是低压。这是高压。压力梯度力单独作用于高到低压。如此所示，由这里的红色箭头表示。所以单独看，如果我们有一些在这个位置的流体，它将朝着L方向运动，沿直线移动到L。方程3.11和3.12右侧的第一个术语，rho fv和减去rho fu，描述了科里奥利力或科里奥利效应。现在，让我通过观察左侧的动画来解释这个。假设我们从太空中俯视地球，这是图片上部的部分。假设在北极有一个黑色的球，观察者是距离北极一些距离的这个红色点。有人站在北极，他们把黑球从中心扔出去，那么从太空中观察，它将沿直线移动，这是因为它遵循牛顿的运动定律，一旦离开人手，就不再受到力的影响。但是对于一个距离北极一定距离的观察者，他们会观察到球实际上朝着其运动的右侧弯曲。它会沿着一条弯曲的路径运动。

So the person stood on the Earth's surface, observing this ball, will think that there's some force acting on this ball which is causing it to bend to the right. This is known as the Coriolis force.

We can also think about this right hand plot of the Earth. If the ball was at the equator, remember the Earth's rotating anti-clockwise when we look down on the North Pole. And the ball would be moving very fast. This is positioned on the equator relative to if the ball were positioned further north. So to an observer from space looking at the Earth and looking at the speed of this ball, he or she observes that the ball is moving very fast, as it is having to rotate around the Earth's circumference within 24 hours. Whereas if the ball were further north of the south, the distance it has to travel in 24 hours is much less.

So the person stood at the equator threw the ball either north or south and as the ball leaves his or her hand, it has that really fast motion associated with it. But as it moves further north the speed of the ground is decreasing. And so it will appear to either bend to the right because it's moving so fast in the north or bend to the left and the south. Either way, in the northern hemisphere, the ball appears to bend to the right of this path. And this is due to the Coriolis force.

If an object is thrown along the equator, then it neither turns left or right. And this is governed by the Coriolis parameter, f, which is two times the rotational angular velocity of the planet multiplied by the sine of the latitude, phi. A latitude is zero at the equator and sine of 0 is 0, so the Coriolis force is 0 at the equator.

Let's go back to slide 7 and think about what this means for our equations of motion. So imagine we've got an object that is moving north. Its only component of velocity is to move north. So, v is positive and u is initially 0. Now, let's think about what happens. So we think about the x direction, v as I've mentioned is positive and non-zero and that results in a positive force. So that means that our object is going to accelerate along the x direction. So, in other words, it's moving from south to north, but it's going to accelerate along the x direction, so it's going to bend. That shows us that the object will bend to the right; moving from north to south, it's going to bend and increase his motion along the x-axis. Whereas for the y direction, u is initially 0. So, there's no such force, at least initially, but that will change as the object starts to bend. So that's just to convince you that an object, if it's moving in the northern hemisphere, it will bend and move to the right and its path.

And now we consider airflow around both high and low-pressure systems. So, as I said before, the pressure gradient force tells us that the Earth will move from an area of high pressure to low pressure. But also, the Coriolis force tells us that the motion of the air will be bent to the right at least in the northern hemisphere. So the air's being pushed away from this high-pressure zone. It will bend to the right. And if air is moving towards this low-pressure zone, shown here, it will also bend to the right. They both bend to the right.

Now, if we join up the arrows, then we get that the motion of the air around a high-pressure system is in a clockwise direction. And if we join up the arrows around this low-pressure system, we get that the motion of the air around the low is anti-clockwise. And we see this in the atmosphere. We see this in cyclones, for instance, and we can see the pattern shown in this cloud banding here from a satellite. So this is a low-pressure centre. And what's happening here is that the clouds are being pulled into this low-pressure system. But the air is bending to the right as it does it. And we can see that pattern in a spiral.

You can also look at other planets. And one such example is Jupiter's red spot. This is an anticyclone but it rotates anti-clockwise again. So that's like a cyclone in the northern hemisphere, but this red spot is actually in the southern hemisphere of Jupiter. So it rotates in the opposite direction. Such a motion like this can be seen to feed off the smaller scale motions, they cascade into it. And because of this feeding off the smaller scale motions, Jupiter's red spot is very persistent and it's actually been observed through telescopes for at least 350 years. So, it's at least 350 years old. You'll model something like this in the practical.

因此，站在地球表面的人观察这个球，会认为有一些作用在这个球上的力，导致它向右弯曲。这被称为科里奥利力。我们还可以考虑地球右侧的图。如果球位于赤道，记住当我们从北极俯视地球时，地球是逆时针旋转的。而且球会移动得很快。这相对于球位于更北位置时在赤道的位置而言。因此，从太空中观察地球并观察这个球的速度的观察者会观察到球移动得很快，因为它必须在24小时内绕地球周长旋转。而如果球在南方更远处，它在24小时内需要移动的距离就少得多。因此，站在赤道的人向北或向南扔球，当球离开他或她的手时，它具有与之相关的非常快的运动。但随着它向北移动，地面的速度减缓。因此，它将似乎向右弯曲，因为在北方移动得很快，或者在南方向左弯曲。无论如何，在北半球，球似乎会向右偏离这条路径。这是由于科里奥利力。如果一个物体沿着赤道被扔出去，它既不会向左也不会向右转。这是由科里奥利参数f来控制的，它是行星的旋转角速度的两倍，乘以纬度phi的正弦。纬度在赤道为零，sin(0)为0，因此赤道上的科里奥利力为0。

让我们回到第7张幻灯片，思考这对我们的运动方程意味着什么。所以想象我们有一个向北移动的物体。它唯一的速度分量是向北移动。因此，v是正的，u最初为0。现在，让我们考虑发生了什么。所以我们考虑x方向，如我之前提到的，v是正的且非零，这导致正向力。这意味着我们的物体将沿x方向加速。换句话说，它是从南到北移动，但它将沿x方向加速，因此它将弯曲。这表明物体将向右弯曲；从北到南移动，它将弯曲并加快沿x轴的运动。而对于y方向，u最初为0。所以，至少最初没有这样的力，但随着物体开始弯曲，情况将发生变化。这只是为了说服你，如果一个物体在北半球移动，它将弯曲并向右移动。现在我们考虑高低压系统周围的气流。所以，如我之前所说，压力梯度力告诉我们地球将从高压区域移动到低压区域。但科里奥利力也告诉我们，空气的运动至少在北半球会向右弯曲。所以空气被推离高压区域，它将向右弯曲。如果空气向着低压区域移动，如图所示，它也将向右弯曲。它们都会向右弯曲。现在，如果我们连接这些箭头，我们就会得到围绕高压系统的气流是顺时针方向的。如果我们连接围绕这个低压系统的箭头，我们会得到围绕低压的气流是逆时针的。我们在大气中看到这一点，例如在气旋中，我们可以在卫星图上看到这种云带的模式。所以这是低压中心。这里发生的是云被拉入这个低压系统。但随着它这样做，空气向右弯曲。我们可以在一个螺旋中看到这个模式。你也可以看其他行星。其中一个例子是木星的红斑。这是一个反气旋，但它再次逆时针旋转。所以这就像是北半球的气旋，但这个红斑实际上在木星的南半球。因此，它以相反的方向旋转。这样的运动可以看作是从较小尺度的运动中汲取能量，它们不断涌入其中。由于从较小尺度的运动中获取能量，木星的红斑非常持久，实际上通过望远镜观察了至少350年。所以，它至少有350年的历史。你将在实际操作中模拟类似的东西。