Level of No Motion (LNM) Pacific: Deep water is uniform, current is weak below 1000m. Atlantic: A level of no motion at 1000-2000m “Slope current”: Relative geostrophic current is zero but absolute current is not. May occurs in deep ocean (barotropic?) Current increases into the deep ocean, unlikely in the real ocean

For a barotropic flow, we have is geostrophic current.  p and ρ surfaces are parallel For a barotropic flow, we have is geostrophic current. Since  Given a barotropic and hydrostatic conditions, and Therefore, And So (≈0 in Boussinesq approximation) The slope of isopycnal is small and undetectable, for V=0.1 m/s, slope~10-6, i.e., 0.1m height change in 100 km

There is no simple relation between the isobars and isopycnals. Relations between isobaric and isopycnal surfaces and currents Baroclinic Flow: and There is no simple relation between the isobars and isopycnals. slope of isobar is proportional to velocity slope of isopycnal is proportional to vertical wind shear. With a barotropic of mass the water may be stationary but with a baroclinic field, having horizontal density gradients, such as situation is not possible In the ocean, the barotropic case is most common in deep water while the baroclinic case is most common in the upper 1000 meters where most of the faster currents occur.

1½ layer flow Simplest case of baroclinic flow: Two layer flow of density ρ1 and ρ2. The sea surface height is η=η(x,y) (In steady state, η=0). The depth of the upper layer is at z=d(x,y). The lower layer is at rest.  For z > d, For z ≤ d, If we assume  The slope of the interface between the two layers (isopycnal) = times the slope of the surface (isobar). The isopycnal slope is opposite in sign to the isobaric slope.

σt A B diff 26.8 50m --- >50m 27.0 130m 280m -150m 27.7 579m 750m -180m Isopycnals are nearly flat at 100m Isobars ascend about 0.13m between A and B for upper 150m Below 100m, isopycnals and isobars slope in opposite directions with 1000 times in size.

Example: sea surface height and thermocline depth

Comments on the geostrophic equation If the ocean is in “real” geostrophy, there is no water parcel acceleration. No other forces acting on the parcel. Current should be steady Present calculation yields only relative currents and the selection of an appropriate level of no motion always presents a problem One is faced with a problem when the selected level of no motion reaches the ocean bottom as the stations get close to shore It only yields mean values between stations which are usually tens of kilometers apart Friction is ignored Geostrophy breaks down near the equator The calculated geostrophic currents will include any long-period transient current

The β-spiral Determining absolute velocity from density field Assumptions: 1) Geostrophic 2) incompressible 3) steady state 2) + 3) 

Use the thermal wind equation with Boussinesq approximation Take into Write u and v to polar format as In the northern hemisphere, if w > 0, the current rotates to the right as we go upward (or to the left as we go downward) 

Take geostrophic equation  If v≠0, w changes with z and can not be zero everywhere. Thus the β effect makes the rotation of the geostrophic flow with depth likely, hence the term “β spiral” use

Consider an isopycnal surface at If we go along this surface in the x-direction  Similarly Take into  

Rewrite the thermal wind relation   Suppose we have derived u’ and v’ based on some reference level If the observations should be error free, two levels would be sufficient Considering the observation errors, particularly noise from time-dependent motions, this equation will not be exactly satisfied. Computationally, a least-square technique is used.