Effects of the Earth’s Rotation C. Chen General Physical Oceanography MAR 555 School for Marine Sciences and Technology Umass-Dartmouth 1 One of the most important physical processes controlling the temporal and spatial variations of biological variables (nutrients, phytoplankton, zooplankton, etc) is the oceanic circulation. Since the circulation exists on the earth, it must be affected by the earth’s rotation. Question: How is the oceanic circulation affected by the earth’s rotation? The Coriolis force! Question: What is the Coriolis force? How is it defined? What is the difference between centrifugal and Coriolis forces? 2 Definition: • The Coriolis force is an apparent force that occurs when the fluid moves on a rotating frame. • The centrifugal force is an apparent force when an object is on a rotation frame. Based on these definitions, we learn that • The centrifugal force can occur when an object is at rest on a rotating frame; •The Coriolis force occurs only when an object is moving relative to the rotating frame. 3 Centrifugal Force Consider a ball of mass m attached to a string spinning around a circle of radius r at a constant angular velocity ω. r ω ω Conditions: 1) The speed of the ball is constant, but its direction is continuously changing; 2) The string acts like a force to pull the ball toward the axis of rotation. 4 Let us assume that the velocity of the ball: V at t V + !V " V = !V V + !V at t + !t ! V = V!" ! V !" d V d" d" r = V , limit !t # 0, = V = V ($ ) !t !t dt dt dt r !V "! V d" V = % r, and = %, dt V Therefore, d V "! = $& 2r dt ω r To keep the ball on the circle track, there must exist an additional force, which has the same magnitude as the centripetal acceleration but in an opposite direction. 5 This force is called “centrifugal force”, and is equals to 2 Fcf = ! r Ω On the earth, the centrifugal force is equal to R Fcf 2 Fcf = ! R ! where Ω is th e angular velocity of the earth’s rotation and R is the position vector from the axis of rotation to be object at a given latitude. 6 The Coriolis Force ω t1 t2 t3 t1 t2 t3 When an objective is moving with respect to a rotating frame, an additional apparent force appears, which tends to change the direction of the motion. The Coriolis force! 7 8 Important Concepts: • Any object on a rotating frame is subject to a centrifugal force no matter whether or not it moves. • The Coriolis force exists only when the object moves on a rotating frame. • The Coriolis force only changes the direction of the motion. • The centrifugal force could accelerate the motion. Questions: How do we define the Coriolis force on the rotating earth? 9 Ω Assume that a fluid parcel moves eastward at a speed of u. Since this parcel moves faster than the earth rotation, so the angular velocity acting on this parcel should be equal to a sum of the angular velocities of the earth and movement of the parcel as R u follows: ! + u / R Therefore, the centrifugal force exerting on this parcel is equal to u F = (! + )2 R cf R Then, u 2!uR u2R F = (! + )2 R = !2R + + cf R R R2 Centrifugal force Too small Coriolis force component 10 (F ) = #2"u sin!, (F ) = #2"u cos! 2"u cos! c y c z R R 2!u Since (F ) << g in the vertical, it can be ignored. R c z 2"u sin! θ Therefore, u Coriolis force on the northern hemisphere Usually, we define that f = 2"sin! as the Coriolis parameter. Fc = fvi ! fuj = ! fk " v 11 Properties 1. The Coriolis force is a three-dimensional force. The vertical component of the Coriolis force is generally ignored in the large-scale ocean study because it is much smaller than gravity. 2. In the northern hemisphere, the Coriolis force acts 90o degree to the right of the current direction, while in the southern hemisphere, it is 90o degree to the left of the current direction. This is a very important concept. 3. The Coriolis force changes with latitude and the amplitude of the currents. At the equator, the Coriolis force equals zero and it increases as the latitude increases towards the poles. 12 Questions: How could the Coriolis effect influence the oceanic circulation? Example 1: Inertial (or near-inertial) motion A fluid parcel Movement direction without the Coriolis effect The Coriolis force t = 0, u = 0, v = vo Discussion: a) 2 2 2 t=π/2f, u + v = vo This is a circle! t=3π/2f, u = vo, v = 0 u =-vo, v= 0 b) Inertial period: 2! T f = f t=π/f, 13 u= 0,, v = -vo The inertial period decreases with latitude, 1) at equator: Tf →∞: no inertial motion because f = 0; o 2) at 30 N: Tf = 23.9 hours o 3) at 45 N, Tf = 17 hours o 4) at 90 N, Tf = 12 hours In the real ocean, an inertial oscillation is usually caused by a sudden change of the wind stress. If you trace a drifter, its trajectory would look like 14 The Louisiana-Texas Shelf Monitoring Sites 15 16 17 Cross-shelf distribution of the variance of the near-inertial currents 18 Clockwise rotation of the wind direction during the cold-frontal passage 19 Example 2: Defining the scale of motion Distance: L Advective time scale: O(L/U) Speed: U For the Coriolis force-induced inertial motion, Inertial time scale: O(1/f) Inertial time scale O(1/ f ) U Ro = = = O( ) Rossby number Advective time scale O(L /U) fL The scale of motion is defined by the magnitude of the Rossby number Ro <<1, Large -scale : Coriolis force is dominant R o ~ 1, Meso -scale : Coriolis force is important and can not be ignored R o >>1, Small-scale Coriolis force can be ignored 20 Example 3: Geostrophic currents FP : The pressure gradient force; Fc : The Coriolis force F (The pressure gradient force) t P Low 0 t1 P0 t2 V P g 1 Fc Fc FP P2 Fc High Fc (The Coriolis force) Coriolis force = Pressure gradient force Geostrophic currents 21 Example 4: Ekman Transport, Currents and Pumping Without the Coriolis force, the water moves following the force direction. With the Coriolis force, (wind stress) t ! s ! s o t1 90o Ekman transport VE Fc Fc Fc ! Coriolis force = Surface wind stress Ekman mass transport: V = s E f ! s Ekman volume transport: VE = "f 22 Consider the wind-driven Ekman currents in the vertical water column ! s V ! s At the surface: 45o v E 45o ! 1 Fc hE VE (Ekman volume transport) ! v ! 2 Below the surface: V ! Fc ! v E surface current o 45 90o V E Current below the surface Clockwise rotates with depth 23 Ekman pumping hE hE 24 Discussion: a) Current profile: The Ekman velocity decreases and rotates clockwise with depth: Ekman spiral. b) The Ekman layer thickness (depth): 2K directly proportional to turbulent viscosity coefficient and inversely h = m E f proportional to the Coriolis parameter. b) The direction of the surface Ekman current: The angle between the wind stress and surface Ekman current v o tan! = E = 1 , ! = 45o is 45 . On the northern hemisphere, the surface Ekman current o uE is 45 on the right of the wind stress. c) The total volume transport: τs ! The volume transport is always 90o to the direction of the 45o v E wind stress. In the northern hemisphere, it is to the right of the wind stress. transport 25 Suggested reading: Chen, C., R. O. Reid, and W. D. Nowlin, 1996. Near-inertial oscillations over the Texas-Louisiana shelf, Journal of Geophysical Research, 101, 3509-3524. 26.