Ocean Modeling - EAS Chapter 2. We model the ocean on a rotating planet

Transcription

1 Ocean Modeling - EAS 8803 We model the ocean on a rotating planet Rotation effects are considered through the Coriolis and Centrifugal Force The Coriolis Force arises because our reference frame (the Earth) is rotating The Coriolis Force is the source of many interesting geophysical processes Chapter 2

2 A rotating framework - The coordinates j J r Y y i x Ωt X I Figure 2-1 Fixed (X, Y ) and rotating (x, y) frameworks of reference Rotating reference x = + X cos Ωt + Y sin Ωt y = X sin Ωt + Y cos Ωt Angular Velocity Fixed reference

3 A rotating framework - The velocity (1st derivative) dx dy = + dx = dx dy cos Ωt + dy sin Ωt + sin Ωt +Ωy {}}{ ΩX sin Ωt + ΩY cos Ωt cos Ωt ΩX cos Ωt ΩY sin Ωt Ωx Relative Velocity change of the coordinate relative to the moving frame u = dx i + dy j Absolute Velocity U = dx I + dy J

4 A rotating framework - The velocity (1st derivative) u dx dy v = + dx = dx U dy cos Ωt + dy sin Ωt + V sin Ωt +Ωy {}}{ ΩX sin Ωt + ΩY cos Ωt cos Ωt ΩX cos Ωt ΩY sin Ωt Ωx Relation between absolute and relative velocity U = u Ωy, V = v + Ωx Ωy Ωx absolute velocity = relative velocity + entraining velocity due to rotation

5 A rotating framework - The acceleration (2nd derivative) du d 2 x 2 = dv d 2 y 2 = ( d 2 X 2 cos Ωt + d2 Y 2 ) sin Ωt ( + 2Ω dx ) cos Ωt dy sin Ωt + V Ω( 2 (X cos Ωt + Y sin Ωt) ( ) ( ) x ( ) ( ) ( d2 X 2 du dv x sin Ωt + d2 Y 2 cos Ωt Ω 2 x ) ( dx 2Ω 2ΩV 2ΩU ) sin Ωt dy cos Ωt + U Ω 2 ( X sin Ωt + Y cos Ωt) ( y Ω 2 y

6 A rotating framework - The acceleration (2nd derivative) du d 2 x 2 = ( d 2 X 2 du cos Ωt + d2 Y 2 ) sin Ωt Ω 2 (X cos Ωt + Y sin Ωt) x ( + 2Ω dx ) cos Ωt dy sin Ωt + V ( ) ( ) Ω 2 x 2ΩV ( Relation between absolute and relative velocity use this equality: U = u Ωy, V = v + Ωx

7 A rotating framework - The acceleration (2nd derivative) du = du 2Ωv Ω2 x dv = dv 2Ωu Ω2 y absolute acceleration = relative acceleration + Coriolis acceleration + Centrifugal acceleration use this equality: U = u Ωy, V = v + Ωx

8 A rotating framework - The acceleration (2nd derivative) define vectors: r = x y u = u v Vector Notation du + 2Ω u + Ω (Ω r) Coriolis acceleration Centrifugal acceleration time derivative in rotating frame: d + Ω

9 Coriolis vs Centrifugal Force du dv = du 2Ωv Ω2 x = dv 2Ωu Ω2 y Centrifugal Force depends only on location Coriolis Force is active only when things move

10 Centrifugal Force is unimportant for motions 22 CENTRIFUGAL FORCE 41 N Ω Gravity Net Local vertical Centrifugal geoid an equipotential surface S Figure 2-2 How the flattening of the rotating earth (grossly exaggerated in this drawing) causes the gravitational and centrifugal forces to combine into a net force aligned with the local vertical, so that equilibrium is reached

11 Free Motion on a rotating frame du = du 2Ωv Ω2 x dv = dv 2Ωu Ω2 y Coriolis Force is active only when things move

12 Free Motion on a rotating frame du 2Ωv = 0, dv The general solution to this system of linear equations is + 2Ωu = 0 u = V sin(ft + φ), v = V cos(ft + φ) f = 2Ω, Inertial Oscillations NOTE: the speed does not change with time yet u and v do change with time! changes in u and v imply change in direction

13 Trajectory of inertial oscillations x = x 0 V f cos(ft + φ) 2 y = y 0 + V f sin(ft + φ), V combine and take the square (x 0, y 0 ) R = V f (x x 0 ) 2 + (y y 0 ) 2 = 1 this is the equation of a circle with radius R = V f ( V f ) 2 2 period of a complete circle is called ninertial a time periodeq T p = 2π/f the plane acc

14 Coriolis acceleration in 3D effective rotation on the sphere Parallel N 2π Ω Ω = 24 hours j y z x k i f = 2Ω sin ϕ rotation of reference frame projection on sphere surface Equator Greenwich ϕ λ Meridian equation of inertial oscillations du fv = 0 dv + fu = 0 S these describe the unforced motion

15 Coriolis acceleration in 3D - observations of inertial motions equation of inertial oscillations du fv = 0 dv + fu = 0 these describe the unforced motion

16 Discretizing the intertial oscillation equations du fv = 0 ũn+1 ũ n t dv + fu = 0 ṽn+1 ṽ n t Euler Method fṽ n = 0 + fũ n = 0 ũ n+1 ũ n t ṽ n+1 ṽ n t Euler Method Implicit fṽ n+1 = 0 + fũ n+1 = 0 when rigth-hand side is at future time