Geostrophic Balance

This is the first post in a series in which I’ll explain key concepts in atmospheric science and geophysical fluid dynamics. I’m starting off with geostrophic balance, one of the two main balances between forces that we assume in geophysical fluid dynamics.

The equations that we usually use to describe flow in the atmosphere — how air moves east/west, north/south, and up/down — are called the primitive equations. We can divide this set of equations into the following four categories: conservation of momentum, conservation of mass, conservation of energy, and the ideal gas law. What we’ll be most interested in here are the conservation of momentum equations. We derive them from Newton’s 2nd law, which tells us that a net force on a system results in acceleration, or a change in velocity/momentum. When we apply this to the atmosphere, we find that changes in the velocity of the air come about in three different ways: gravity, friction (which we’ll mostly ignore), and pressure gradients, which are changes in pressure as we move in some direction.

This sounds all well and good, but there’s one problem. The earth rotates, and the conservation of momentum equations described above consider velocities in a non-rotating reference frame. But we live on earth and rotate with it, so we want to measure winds in a reference frame that also rotates with the earth. When we move to a rotating frame, we get the equation

\frac{D\vec{u}}{Dt} = \vec{g} + \vec{f} - \frac{1}{\rho} \vec{\nabla} p - 2\vec{\Omega} \times \vec{u} - \vec{\Omega} \times (\vec{\Omega}\times \vec{r}),

where \vec{u} is the fluid velocity (the winds), \rho is the density, p is the pressure, \vec{g} is gravity, \vec{f} is friction, \vec{\Omega} is the angular velocity of the earth, and \vec{r} is position. That derivative uses capital Ds because it’s a material derivative, \frac{D}{Dt} = \frac{\partial}{\partial t} + \vec{u} \cdot \vec{\nabla}. In some other areas of physics this is called a total derivative. It’s using the chain rule to capture all the variation in time of a quantity — how the quantity directly changes in time and how it changes with position while position changes with time.

If we look at the righthand side of that equation, the first three terms are exactly what we had before, but now there are two new terms. The last term is centrifugal force, and we’ll combine it with gravity to get a corrected gravity. That correction turns out to be pretty small. The second-to-last term, though, is the Coriolis effect, and it’s essential to studying rotating fluids. We can make all kinds of approximations, but the Coriolis term will stick around, especially at large scales.

When we write out our conservation of momentum equations in spherical coordinates (longitude, latitude, and distance from the center of the earth), the Coriolis term appears in two of the three equations, those governing zonal (east/west) and meridional (north/south) motion. We’ll use u  to represent zonal velocity and v  to represent meridional velocity. The Coriolis terms in the u and v equations respectively look like -(2\Omega \sin \phi)v and (2\Omega \sin \phi)u. Here \Omega is the magnitude of the earth’s angular velocity and \phi is latitude. We notice from looking at these two that the term that seems to matter is 2\Omega \sin \phi.

Spherical coordinates are a natural way to write these equations because the earth is close to spherical, but we often prefer to work in a Cartesian frame. At scales that aren’t too large, we can approximate by thinking of looking at a plane that’s tangent to the earth’s surface at some point. If we don’t go too far from that point, the plane represents positions on the sphere well enough. On this plane, our horizontal coordinates will be x and y instead of longitude and latitude. The conversion from \phi to y is

y=a(\phi - \phi_0),

where \phi_0 is the latitude at the point where the plane is tangent to the earth and a is the earth’s radius. To include the Coriolis terms in our equations, we want to rewrite 2\Omega \sin \phi in terms of y, so we use the first couple of terms of a Taylor series expansion:

2\Omega \sin \phi \approx 2\Omega (\sin \phi_0 + \cos \phi_0(\phi - \phi_0)) = 2\Omega \sin \phi_0 + \frac{2\Omega}{a} y \cos \phi_0.

This gives us a linear function in y. We’ll call the Coriolis coefficient f and write this as f = f_0 + \beta y, where 2\Omega \sin \phi_0 = f_0 and \frac{2\Omega}{a}\cos \phi_0 = \beta. If the scales we’re interested in are small in the north-south direction, then we’ll truncate this to just f \approx f_0. This is called the f-plane approximation. If we keep the second term, we call that the \mathbf{\beta}-plane approximation.

Using all of this and neglecting friction, our conservation of momentum equations involving zonal and meridional motion are

\frac{Du}{Dt} - fv= -\frac{1}{\rho} \frac{\partial p}{\partial x},

\frac{Dv}{Dt} + fu = -\frac{1}{\rho} \frac{\partial p}{\partial y}.

In the atmosphere, we often have significant pressure gradients in the horizontal directions. (You may have seen a weather map with high and low pressures marked.) For the equations above to be true, the terms on the lefthand side need to balance those pressure gradients. To determine which of the two terms are doing more of the work or whether both are contributing in similar amounts, we look at the ratio of \frac{Du}{Dt} to fv. (The same analysis would apply to the second equation.)

Let U be our velocity scale and L be our length scale. Then we can write our time scale T= \frac{L}{U}. This gives us

\left|\frac{fu}{Du/Dt}\right| \sim \frac{f_0 U}{U \frac{U}{L}} = \frac{f_0 L}{U}.

That quantity is dimensionless, and we call it the Rossby number, Ro. In the mid-latitudes (30-60 degrees S or N), the Rossby number is typically on the order of 10. This is a large Rossby number, so we can conclude that the Coriolis terms dominate the material derivatives; it’s the Coriolis terms that are balancing the pressure gradients. Using that approximation, we find

fv = \frac{1}{\rho} \frac{\partial p}{\partial x},

fu = -\frac{1}{\rho} \frac{\partial p}{\partial y}.

These are the geostrophic equations, also known as geostrophic balance.

There are a few things to glean from these equations. First of all, they’re time independent; geostrophy is an equilibrium of sorts. Second, what these mean is that the Coriolis force is deflecting the flow so that it moves along lines of constant pressure. We would generally expect flow to move down the pressure gradient, just like you would expect water to flow to the saltier side of a membrane to neutralize a concentration gradient. But the Coriolis force is balancing the pressure gradient, so it’s deflecting the flow to move along constant-pressure lines instead of across them. In the Northern Hemisphere this deflection is to the right, and in the Southern Hemisphere this deflection is to the left.

If we make a couple more approximations, we can reach two more key conclusions. If we make the f-plane approximation, taking f=f_0, then we call the resulting u and v the geostrophic winds, u_g and v_g. The fact that the equation is time-invariant means that these are time-invariant winds. When we want to make predictions about weather, we generally separate the winds into their geostrophic parts (which don’t change with time) and their ageostrophic parts (which do evolve in time).

If we further assume that the density is constant (good approximation in the ocean, sketchy but sometimes useful in the atmosphere), then we can use the two equations to find that

\frac{\partial v_g}{\partial y} + \frac{\partial u_g}{\partial x} = 0.

But remember from the very beginning that we assume conservation of mass. What conservation of mass tells us is that

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0,

where w is the vertical velocity and z is our vertical coordinate. But in geostrophy with constant density, the first two terms alone give us 0! So then \frac{\partial w}{\partial z} = 0, telling us that vertical velocity is constant with height. Further, the vertical velocity at the ground (or at the bottom of the ocean) must be 0, so for all heights, the vertical velocity is 0. In other words, the rotation of the earth — which is what gives us the Coriolis term — makes our flow two-dimensional. Even if we don’t have constant density, this approximation is pretty good. In practice, on large horizontal scales, the vertical velocity is much smaller than the horizontal velocities. This is convenient because 2D flow is a lot easier to work with than 3D flow, and far more is known about it.

So that’s geostrophic balance! It’s time-invariant horizontal flow along lines of constant pressure, it arises from the rotation of the earth, and it tells us that we can approximate our system as two-dimensional.

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