Simple Oceanic Motions Influenced by Rotation
1. The Coriolis Force: Accelerated and Inertial Coordinate Systems
Consider the form of the Navier-Stokes equations in an accelerated frame of reference; i.e., a rotating reference frame on the surface of the earth. Momentum conservation is:
(1)
Where: Uf is the velocity vector in a fixed or non-accelerated coordinate system. This expression is directly useful in three circumstances:
§ When the motion considered is small-scale and fast, so that rotation effects are small relative to other accelerations
§ When the frame is located on the axis of motion, at one of the planet’s two poles; and
§ When the planet isn’t rotating.
Obviously, it is necessary to consider the consequences of working in a rotating coordinate system.
Points about rotating coordinate systems and the origin of the Coriolis force:
§ The difference between a rotating and a fixed (relative to the center of gravity of the earth) coordinate system arises because a rotating coordinate system is accelerated, causing fictitious forces.
§ The Coriolis force is analogous to the centrifugal acceleration that one experiences riding in a car, when the car goes around a corner.
§ The apparent force in the car can be understood in terms of Newton's 2nd Law; F= MA.
If there is no force F, then there is no acceleration A, so speed remains constant.
§ The car exerts a centripedal force when it corners that balances the apparent centrifugal force; i.e., the tendency of the passenger’s body to continue in a straight line.
For a moving fluid on the surface of the earth, the rotating earth exerts the apparent force:
Figure 1: In the northern hemisphere, a body moving northward has an excess of angular momentum, because it comes from a position that has a greater radius – it moves to the right of north. If a body moves sourthward, then it has less angular momentum than its new surroundings and again moves right of southward. The movement is always to the right of the initial direction in the northern hemisphere and to the left in the southern hemisphere. Above, the dotted vector shows the initial velocity and the solid vector shows the actual motion.
Quantifying Acceleration in a Rotating Frame
Figure 1 is not the whole story because east-west motion is also affected, not just motion in a north south plane. Lets see why. To translate (1) into a rotating coordinate system, we need a rule for expressing the difference between a time-derivative in an accelerated (non-inertial) reference frame and the same derivative in a motionless frame. The transformation between the two is:
(2)
where W is a vector along the axis of rotation whose length is proportional to the speed of rotation, subscript f refers to motion in a fixed coordinate system, subscript r refers to motion in the rotating coordinate system, and vec is an arbitrary vector. If vec is the position vector x, then (2) expresses the difference in velocities between rotating and fixed coordinate systems:
(3)
Points about (3)
§ The rate of change in the rotating coordinate system is that in the fixed coordinate system plus a correction related to the rate of rotation; the origin from which xr and xf are measured is the center of the earth (point O in Figure 2).
§ This theorem, given without proof, it is plausible from the geometry of the situation (Figure 2).
§ Points on the equator have maximal angular momentum and velocity associated with rotation of the earth
§ Points at the poles have no angular momentum
§ Intermediate points have a velocity proportional to sin j, where j is co-latitude (90°-latitude) and the angle between the rotation vector W and xr:
Figure 2: W x xr vanishes if the point on the surface of the earth xr is along the axis of rotation W; i.e., if xr is at one of the poles. W x xr is at a maximum if the point xr is on the equation, so that the angle j between xr and W is 90°. At latitudes between the equator and the pole, W x xr has an intermediate value proportional to sin j where j is co-latitude. (Picture from A. E. Gill, Atmosphere-Ocean Dynamics, p. 73).
(1) defines a relationship for velocity dx/dt. But (1) applies to any vector, so it can be applied recursively to the velocity to derive a relationship for accelerations. If it is applied to the results of (1):
(4)
where: U º dx/dt, and R is the radius of point xr from W, i.e., the distance from the axis of rotation (not the distance from the center of the earth xr).
The meaning of (4) is that the acceleration of a point on the surface of the rotating earth (expressed in a rotating coordinate system) contains two "extra" terms:
· 2W x Ur, commonly known as the Coriolis force, and
· A centripedal acceleration W2 R that will be combined into the gravitational acceleration g.
Including these "apparent" or fictitious forces in the equations of motion is the price that must be paid for doing all our analyses in an accelerated frame of reference fixed to the surface of the rotating earth, as opposed to one stationary relative to the distant stars. While this may seem bizarre and unintuitive, it is much better than the other alternative. Consider working in a fixed reference frame -- the boundaries of any water body (e.g., the ocean shore) would be constantly moving.
We still need to:
§ Consider the magnitude of W2 R and its effects on the apparent gravity at any on the earth.
§ Write out the component equations of (1) including 2W x Ur, and
Modification of gravitational acceleration g = 9.81 ms-2 by the earth's rotation –
§ Objects do not spontaneously fly of the earth at the equator – the effect of W2 R is small and depends on R and the rate of rotation of the earth (W = 7.29 x 10-5 rad s-1).
§ The maximum possible value of R is at the equator; R = 6378 km, where W2 R = 0.03389 ms-2. vanishes at the poles.
§ W2 R/g £ 0.00346, indicating why objects do not spontaneously leave the earth.
It is customary to consider at the same time as centrifugal effects on g, the effects of flattening of earth at its poles; i.e., its slight "pear shape".
The effective g @ 9.8 ms-2 used in oceanographic problems is a function of latitude (longitude, if one wishes to use the best modern geoid, as is necessary in satellite oceanography). This positional dependence of g is, however, small relative to other approximations used in fluid mechanics (e.g., the neglect of horizontal turbulent diffusion) and can be neglected for our purposes.
Figure 3: The local vertical defined by the sum
of gravitational acceleration and the centrifugal
force does not point exactly to the center of earth.
The non-spheroidal shape of the earth also con-
tributes to the orientation of the effective g. (From
B. Cushman-Roisin, Introduction to Geophysical
Fluid Dynamics, p.20). f + q = 90º
Expression of the Coriolis force 2W x Ur in terms of its components is easy:
(5)
Approximations:
§ The final line in (5) neglects W as small relative to U and V in the x-and y-component equations. That is, the 2Wx and 2Wy (2Wx = 2Wy =2W cos q) parts of the Coriolis force in the x and y equations of motion.
§ Neglect of the Coriolis force in the vertical or z equation of motion is another aspect of the Boussinesq approximation - the vertical acceleration is assumed small, regardless of coordinate system.
Functional Equations of Motion in a Rotating Coordinate System --
Now we define the vertical component of 2W, or 2Wz º f º 2W sin q, where q is latitude – A typical value of f at mid-latitudes is 2W sin 45° @ 1.03 x10-4 s-1 (Figure 4)
The final working form of the momentum equation (1) in a rotating frame is then:
(6)
Figure 4: The vertical component 2Wz of 2W vanishes at the equator and is at a maximum at the pole; it varies with sin q, where q is latitude.
(6) can be written in terms of the component equations, making the Boussinesq approximation:
(9)
The continuity equation and scalar conservation equations for salt and temperature are unaffected by the change between coordinate systems. Do you understand why?
Some final points:
· f is not a constant; f = 2W sin q, where q is latitude. If one orients a right-handed coordinate system on the rotating earth such that the x-axis points east and the y-axis points north, then east-west motion leaves f unchanged, but north-south motion changes f; i.e., f is a function of y. This means that taking a y-derivative of the x or y-components of the momentum equations introduces a ¶f/¶y term. For most motions of interest to us, the scale of the motion is too small to worry about this correction, but it is vital for understanding large scale ocean and atmospheric circulation. There is a whole class of Rossby waves for which ¶f/¶y (not gravity) is the restoring force that makes the wave possible.
· A common form of ocean-atmosphere slang: "meridional circulation" means that occurring in a north-south oriented plane along lines of constant longitude. "Zonal circulation" means that occurring parallel to the equator along lines of constant latitude.
· The Coriolis terms neglected in (9) (compare (9) to (5)) are called the “non-traditional Coriolis terms”. These terms actually make a difference in processes like ocean tides. Their effects are just beginning to be studied.
2. Geostophic flow
Important ideas about Geostrophy:
§ Most large-scale flows in the interior of the ocean are almost geostrophic; i.e., the pressure gradient is balanced by the Coriolis "force" (which arises solely because of our fixed perspective on a rotating earth), and all other forces are small.
§ Most of modern large-scale oceanography focuses on the part of the flow that is not in geostrophic balance, because that is what makes important things happen. If large scale motions were completely geostrophic:
o We wouldn’t get rain out of low pressure systems.
o Summer high pressure wouldn’t bring dry weather.
o All oceanic motions would be steady.
§ Geostrophy was at the frontiers of oceanography a century ago.
Where does the concept of a geostrophic momentum balance apply:
· It applies to a time-average view of the interior of the ocean or atmosphere. THIS IS MOST OF THE WORLD OCEAN AND A LARGE PART OF THE ATMOSPHERE
· It applies away from the surface (where winds and waves act directly) and the sea bed, where friction is important.
· The realm of applicability is then perhaps below 10-50 m from the surface of the ocean, and somewhere between a few cm to ~100m above the seabed on time scales of days to months.
· Geostrophy excludes all time variations in properties -- waves of any sort, storms and any other time-varying motion.
· A century ago, the realm defined here was largely inaccessible, because of the difficulty of making measurements, and because measuring a time average implies having instruments capable of making measurements over a period of months to a year a more. Aside from tide gauges and thermometers, such instruments simply did not exist a century ago. Scientists of the time had very little intuition about such large-scale flows involving an unfamiliar fictitious force.
Defining a geostrophic flow balance will be carried out here as a scaling exercise, but that is not how it was done historically. A century ago, there was no way to know anything about turbulence in the interior of the ocean away from the surface. Measurements (temperature and salinity determined via water bottles) showed short-term variations that were eventually attributed to internal waves, but could not reveal the presence or absence of friction. Thus, an intellectual leap was required, which was eventually justified by analogy to the more easily observed atmosphere.
One approach is simply to assume that friction is absent, but a better argument is available. First we assume we are interested in steady motions only, justifying the interest of this approach through a scaling argument. Friction is then excluded by the argument that flow in the interior of the ocean does not have any obvious driving force. If friction were important, then this motion could not be steady. So the assumption of steady flow is equivalent to assuming inviscid motion. Actually, this argument is, in general, wrong – it turns out that the non-linear terms in the equations of motion can generate nearly steady motion through interaction of waves (either tides or large-scale waves in the ocean that are analogous to weather systems). In the same way, a series of storms in the atmosphere can result in longer term changes in temperature or precipitation; e.g., a wet cold year. Such non-linear motions are not as large (on average) as geostrophic flow, and weren't discovered/understood until after WWII. Fortunately, as above, large-scale oceanic motions are approximately geostrophic, so the argument below works, approximately.[1]
Consider now scaling the x-momentum equation for the flow along a continental shelf oriented in a N-S direction. The y-momentum equation is very similar (you can do it yourself). Note that we assume that all large-scale flows are hydrostatic, so we don't have to do the z-equation again. Also remember that scales correspond to upper limits, to see which terms might have to be included under the most general circumstances. Define the following scales:
U » V = 0.5 ms-1 horizontal velocity field
Lx » 105 m width of the continental shelf
Ly » 5 x105 m along-shelf scale
f » 10-4 s-1 for mid latitudes
T » 5x 106 s or ~5.5 d a typical scale for weather induced changes
Note that W scales as U H/Lx. Scaling the pressure term is a problem -- we don't apriori know how to do that. But if the flow is hydrostatic, the pressure scale is: