Unit 6 – Part I: Geostrophy Introductory Physical Oceanography (MAR 555) - Fall 2009 Miles A. Sundermeyer Unit 6 – Part I: Geostrophy Assigned Reading: OC 3.3 and IPO Chapter 10

Key Concepts: Recap rotating (non-inertial) reference frames Motions on a rotating Earth Momentum equations a.) Geostrophic Balance: b.) Hydrostatic Balance: c.) Thermal Wind:

Recap: Coriolis Force apparent trajectory in rotating ref. frame actual trajectory in inertial ref frame

Recap: Coriolis Force http://paoc.mit.edu/labweb/lab5/gfd_v.htm

Inertial Circles y y impulse impulse x x velocity position

“Balance” of forces in inertial motion Centrifugal force Coriolis Force In Northern Hemisphere What’s wrong with this?

Steady Forced Inertial Motions forcing velocity x y x position

Steady Pressure Driven Motions – a.k.a. Geostrophy

Steady Pressure Driven Motions – a.k.a. Geostrophy http://www.newmediastudio.org/DataDiscovery/Hurr_ED_Center/Hurr_Structure_Energetics/Spiral_Winds/Spiral_Winds.html

Scaling the (u/v)-momentum equations Basin Scale We have no easy basis for scaling the horizontal PG Acc + Adv = PG + Coriolis+ Friction - Pressure Gradient must be Order 1 to Balance Coriolis 10

Scaling the w-momentum equation Acc + Adv + Adv + Adv = PG + Grav + Diff + Diff + Diff We can ignore all but the pressure gradient and gravity – AT THESE SCALES Note: Advection terms all same order of magnitude Friction terms all same order of magnitude 11

Geostrophy / Thermal Wind Geostrophic equation: Hydrostatic equation: Eliminating pressure implies: Similarly, for u: Thermal Wind Relations

Geostrophy / Thermal Wind (cont’d) Example: Constant density layers Geostrophy implies: r0 r1 r2 r3 e.g., see OC Fig 3.20 1 2 3 x z y Dz1 Dz2 Dz3 Dx  Thus the contributions to DP at each level are: level DP v 1 (r1-r0)gDz1=DrgDz pos 2 -(r2-r1)gDz2 =-DrgDz 0/neg 3 (r3-r2)gDz3 =DrgDz

Geostrophy / Thermal Wind (cont’d) Example: Constant density layers Alternatively, Thermal Wind implies: r0 r1 r2 r3 e.g., see OC Fig 3.20 1 2 3 x z y Dz1 Dz2 Dz3 Dx r0 1 Dz1 r1 Thus associated with each isopycnal is: Dz2 2 r2 interface r/x v/z v 1 pos neg decreases with +z 2 increases with +z 3 3 Dz3 r3 Dx x z y

Geostrophy / Thermal Wind (cont’d) Example: Constant density layers layer / interface Geostrophic v Thermal Wind v 1 pos decreases with +z 2 0/neg increases with +z 3 r0 r1 r2 r3 x z y v(z)

Geostrophy / Thermal Wind (cont’d) Example: Constant density layers r0 r1 r2 r3 x z y v(z)

Geostrophy / Thermal Wind Geostrophic equation: Thermal Wind Relations:

Dynamic Height (E.g., see OC 3.3.4; Stewart 10.4) Specific Volume: Specific Volume Anomaly: NOTE: Larger d corresponds to lower density From hydrostatic equation: A B Po rA rB Implies that lower density requires greater height of water column above

Dynamic Height (cont’d) Change in dynamic height: Momentum Equation (geostrophic balance): Hydrostatic Equation: A B Po rA rB NOTE: 1 Dynamic meter = 1 geometric m / 9.8

Dynamic Height (cont’d) Example: Global dynamic topography Introduction to Physical Oceanography, Stuart, Fig 10.2 Ocean Circulation, Open University, Fig 3.21

Dynamic Height (cont’d) Example: Global dynamic topography Ocean Circulation, Open University, Fig 3.21 Fig 3.22 Topographic map of the mean sea-surface (i.e., the marine geoid), as determined using a satellite-borne radar altimeter. The mean sea-surface topography reflects the topography of the sea-floor rather than geostrophic current flow, as the effect of the latter is about two orders of magnitude smaller, even in regions of strong current flow.

Baroclinic vs. Barotropic Ocean Circulation, Open University, Fig 3.11 Barotropic: levels of constant pressure are parallel to surfaces of constant density. Baroclinic: levels of constant pressure are inclined to surfaces of constant density. Ocean Circulation, Open University, Fig 3.15

Computing Currents from Hydrographic Observations Slopes of sea surface O(1:105 to 1:108) Isopycnal slopes several hundred times larger Thermal wind only tells you vertical shear, not absolute velocity Level of “no” or “known” motion Small temporal/spatial variations in density confound geostrophic flow estimates Large-scale measurements allow estimates of average velocities Can only compute geostrophic velocities normal to hydrographic transects

http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/fw/crls.rxml