Balanced Flow: Large-Scale Balances Geostrophic Motion Now go back to the equations of motion in a rotating framework

Geostrophic Balance

A very powerful constraint on large-scale flows

But what happens near the equator?

Geostrophic flow is horizontally nondivergent! At a boundary But the atmosphere is not incompressible

So we head back to pressure coordinates! Geostrophic Wind in pressure coordinates

Highs and Lows in Synoptic Charts Gradient wind balance Balanced flow in the radial-inflow experiment Angular Momentum

Magnified at small r

Cyclostrophic and Geostrophic limits of Gradient Wind Balance The Taylor-Proudman Theorem

Vertical Component of

is of

If

is the gradient along

The Taylor-Proudman Theorem Taylor-Proudman Theorem

Taylor Columns

The Thermal Wind Equation

Geostrophic flow should increase with height True in the real world But the atosphere is ot really arotropi……so

Consider water -

So thermal wind is just geostrophy and hydrostatic balance!

Analogous to - Thermal Winds in the Lab In Spherical Coordinates The Thermal Wind Equation and the Taylor- Proudman Theorem

How does the fluid adjust on large scales when gravitational pull downward is counterbalanced by the rigidity of the Taylor columns? This is a general statement of the thermal wind Reduces to

Because is parallel to But…………………. for a Barolii Fluid -

If

is Baroclinicity If and

Then (7 – 20) is the same as

This is why temperature surfaces can maintain the slopes despite gravity – Earth’s rotatio a alae gravity Cylinder Collapse under Gravity and Rotation

Theory following Margules Margules relation

Mutual Adjustment of Velocity and Pressure Rossby Adjustment Problem

Azimuthal speed is given by Assume

Combine 7-21 and

Slumping will continue till vertical shear is enough to satisfy 7-21 and this will occur when H is the vertical scale of motion and

Is the horizontal length scale – Rossby Radius of Deformation Rossby radius of deformation is the scale at which the effects of rotation become comparable to those of stratification

On scales smaller than the Rossby radius, pressure adjusts to the velocity field whereas at scales much greater than , velocity adjusts to the pressure.

is ~1000 Km in the atmosphere and ~30 Km in the ocean Thermal Wind in Pressure Coordinate

For compressible fluid, go to pressure coordinate

Take the p-derivative of the x-component of

Height contours on pressure surfaces are streamlines for geostrophi flow….teperature otours o pressure surfaces are streamlines for the thermal wid shear……. Pressure contours on constant height surfaces are streamlines of geostrophic flow. But it has density in it.

T contours are streamlines of geostrophic shear,

Subgeostrophic flow – The Ekman Layer For

But if friction term is not small then - Flow is subgeostrophic The ageostrophic component is always directed to the right of in the northern hemisphere. Frictionally induced cross-isobaric flow Ageostrophic flow in atmospheric highs and lows A simple model of winds in the Ekman layer

Assume that the x-axis is directed along the isobars and surface stress drops exponentially over Then

yields is weaker than its geostrophic value and

Where do you expect the winds to be closer to its geostrophic value – land or ocean? Vertical motion induced by Ekman layer

Planetary-scale ageostrophic flow