Oct 14: The upper ocean response to winds

 

 

Wind stress and drag coefficients

 

 

Wind blowing over the sea surface exerts a stress on the ocean that imparts momentum. This drag force at the surface slows the wind speed and forms a boundary layer in the atmosphere. The precise details of how the atmosphere and ocean interact to exchange momentum is complicated by the stratification or stability of the atmospheric boundary layer, the presence of waves, wave breaking, and a host of other processes.

 

For practical applications in oceanography, it’s enough for us to use an empirical formula to calculate wind stress from wind speed.

 

 

where

 

U10  = wind speed at 10 m above the sea surface

= 1.22 kg m-3

          CD = dimensionless drag coefficient

a typical value might be 0.0013

 

this gives  in units of N m-2, or Pascals (Pa).

 

 

A popular formula for a neutrally stable boundary layer is that of Large and Pond (1981), J. Phys. Oceanog., 11, 324-336.

 

CD = 1.2 x 10-3                                           for     4 < U10 < 11 m s-1

CD = 10-3 (0.49 + 0.065 U10)     for     11 < U10 < 25 m s-1

 

Example:  U10 = 8 m s-1      = 1.2 x 10-3 * 1.22 * 64 = 0.0937 Pa

 

 

Typical oceanic values of wind stress are around 0.1 Pa.

 

Because wind stress is a quadratic function of wind speed, gusty winds produce larger stresses than would steady winds of the same average speed.

 

Stormy regions, such as the Southern Ocean, have particularly high mean wind stress.

 

In practice, we often have observations from instruments located as some height other than 10 m above the sea surface. However, it is meteorological convention to report wind speeds as equivalent 10 m values by using the log layer theory (Monin-Obukov theory) to adjust the observed wind speed to that which would have been observed if the anemometer were at 10 m. This adjustment can be a source of error.

 

At low wind speeds there is considerable uncertainty about the correct parameterization of drag coefficient, and this is an active research problem.

 

Direct estimates of drag coefficients at low wind speed vary wildly, and there appear to be more factors involved that simply the wind speed profile through the logarithmic atmospheric boundary layer. These include:

 

·        whether the log layer assumption is valid at low wind speed

·        the presence of waves and swell

·        surfactants

·        meso-scale variability affecting stability of boundary layer

 

 

We have now introduced a complete description of the various forces that act on a fluid and govern its motion.

 

 

 

 

 

which include the forces:

 

·        pressure gradients

·        gravity

·        stresses (viscous and turbulent)

·        Coriolis

 

 

The sum of these forces starts or keeps a fluid in motion by producing a net acceleration of the fluid.

 

The acceleration is comprised of a local time rate of change, but also changes following the fluid. So even in steady flow, where nothing changes with time, fluid may gain or lose momentum as it flows along.

 

This fluid motion occurs subject to constraints on the conservation of mass, and since motion transports temperature and salt which affect density and therefore pressure, transport processes are directly linked to the dynamics of the flow itself.

 

Understanding fluid motion requires an understanding of the balances of forces.

 

We already have some intuition for how fluid behaves, and this stems from understanding simple force balances:

 

·        water accelerates when you pour it      (gravity : acceleration)

·        sticky fluids pour at a steady rate        (gravity : friction)

·        water pistol                                        (pressure : acceleration)

 

Geophysical fluid dynamics is the study of fluid flow on scales large enough that the fact we are on a rotating planet is of fundamental importance.

 

The effects of the Coriolis force are often counter-intuitive – but this is just a matter of us developing a new intuition.

 

 

 

Effects of the Earth’s rotation on oceanic motion

 

Chapter 6 Knauss

 

In discussing friction we considered the Reynolds number

 

 

Re = ratio of nonlinear momentum advection (or inertia) terms to viscous terms.

 

The Rossby number we’ve met in previous lectures is

 

Ro = ratio of  inertia terms (a.k.a. centrifugal acceleration) to the Coriolis acceleration:

 

         

 

where          U is a velocity scale

L a length

f is the Coriolis parameter  2W sin (latitude)

W = 2p/(86164 seconds) = 7.29 x 10-4 s-1

typical f at latitude 45o is  10-4 s-1

 

 

Let’s consider the dynamics that result for a very simple force balance for a flow with Ro ~ 1.

 

Inertial motion

 

Shipwrecked: In a wide expansive ocean with no coastlines and no gradients in anything in any direction

 

·        no pressure gradient

·        no friction

·        no spatial gradients in velocity

·        the wind has stopped blowing, but the fluid is in motion

 

All that is left is the balance between inertia and Coriolis

 

         

 

These equations can be solved by differentiating one with respect to time, and substituting in the other to give a single equation for v (or u).

 

         

 

Applied mathematicians, engineers and musicians will recognize this as a wave equation with the simple solution

 

         

 

which is readily verified by substitution into the equation. It follows that

 

 

The magnitude of the velocity is

 

         

 

which is constant in time.

 

The water is always going the exact same speed. This speed is the speed the water had attained when the wind (or whatever started it going) stopped.

 

This fits with our knowledge of Newton’s Laws of Motion that in the absence of any force to change the momentum, the fluid will carry on without accelerating or decelerating.

 

The peculiar thing about this is that the direction keeps changing.

 

These velocity components describe motion in a circle of radius

 

 

The direction of motion is clockwise for f>0 (northern hemisphere) and counter-clockwise for f<0 (southern hemisphere).

 

(We also describe the direction as anti-cyclonic regardless of the sign of f for reasons that will be become apparent when we discuss the direction of rotation of cyclones and anticyclones in the pressure field. Anti-cyclonic implies clockwise in the northern hemisphere AND counter-clockwise in the southern hemisphere).

 

 

For example, water in motion with a speed of U = 0.5 m s-1, at latitude 42oN where f = 10-4 s-1, describes a circle of radius

 

          r = 0.5/10-4 = 5000 m = 5 km.  

 

The period of the motion (once around the circle) is

 

           

 

which we call the “inertial” period.

 

Notice that it changes with latitude.

 

·        At 60o latitude the inertial period is 13.8 hours.

·        At 45o latitude the inertial period is 16.9 hours.

·        At 30o latitude the inertial period is 23.9 hours.

 

 

Although we assumed no spatial gradients in the flow, which would seem to make specifying a characteristic length scale in the Rossby number rather ill-posed, we see that a natural length scale (the radius of the inertial circle) arises in the solution.

 

The Rossby number for this inertial oscillation phenomenon is:

 

          Ro = = 1.

         

This says the inertia forces are of the same size as Coriolis, which is hardly surprising since this is precisely the simple balance we assumed at the start.

 

These so-called “inertial oscillations” are often observed in the ocean in situations such as the response to the passage of an abrupt storm. They will be apparent in current meter observations of u(t) and v(t) at a fixed point, and also show in the trajectories of drifting buoys.

 

 

 

 

Stewart: Figure 9.1 Inertial currents in the North Pacific in October 1987 (days 275–300) measured by holey-sock drifting buoy drogued at a depth of 15m. Positions were observed 10–12 times per day by the Argos system on NOAA polar-orbiting weather satellites and interpolated to positions every three hours. The largest currents were generated by a storm on day 277. Note these are not individual eddies. The entire surface is rotating. A drogue placed anywhere in the region would have the same circular motion. From van Meurs (1998).

 

 

Further reading :

Stewart chapter 9

Knauss chapter 6