Vertical Coordinates

Vertical coordinates Physical climatology and climate analysis with numerical models Lecture : Prof . Shimon Krishak Presented by : Avi Luvchik

Layout 1. 2. 3. 4. 5. 6. 7. 8. 9.

Vertical coordinate Altitude coordinate Shortly Pressure coordinate Conversion from altitude to pressure Sigma coordinate Conversion from pressure to sigma Eta coordinate The differences Summary

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Vertical coordinate A model's vertical structure is as important in defining the model's behavior as the horizontal configuration and model type. Proper depiction of the vertical structure of the atmosphere requires selection of an appropriate vertical coordinate and sufficient vertical resolution.

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Altitude coordinate n

In the altitude (z) coordinate, layer top and bottom are defined as surface of constant altitude and pressure varies in the x and y direction along these surface. Altitude is an independent variable, and pressure is a dependent variable. This coordinate causes problem when surface elevation are not uniform.

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Altitude coordinate

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Pressure coordinate n

In the pressure coordinate, layer top and bottom are defines as surface of constant pressure. Since altitude is function of pressure in the x and y direction, pressure is independent variable and altitude is dependent variable. The pressure coordinate does not erase problem associated with surface topography.

n

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Pressure coordinate

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Conversion from altitude to pressure q 2 − q3 q1 − q3  p 2 − p1  q1 − q 2   = +  x 2 − x1 x 2 − x1  x 2 − x1  p1 − p 2 

The relationship is an exact equivalence as.

x 2 − x1 → 0 and p1 − p 2 → 0 q 2 − q3  ∂q    =  ∂z  z x 2 − x1

q1 − q3  ∂q    =  ∂x  p x 2 − x1

p 2 − p1  ∂p a    = x 2 − x1  ∂x  z

 ∂q   ∂p a

 q1 − q 2  =  x p1 − p 2

Substituting these terms into Eq. 1 gives

 ∂p a   ∂q  ∂q   ∂q  = +         ∂x  z  ∂x  p  ∂x  z  ∂p a

  x 8

Which generalized for any variable as  ∂p   ∂ ∂  ∂    =   +  a    ∂x  z  ∂x  p  ∂x  z  ∂p a

  x

A similar equation can be written for ydirection. Combining the x and y equations in a vector from the horizontal gradient conversion from Cartesian altitude to Cartesian pressure as.

∂ ∇z = ∇p + ∇z ( p a ) ∂p a

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Sigma coordinate n

Because of the complication of the bottom boundary condition Philips (1957) introduced to “normalized pressure” or sigma coordinate.

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Sigma coordinate ( σ ) n

the sigma coordinate is defined by

n n 

n

where p is the air pressure, and the subscripts s and t refer to the surface and top of the model.

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The advantages of the σ coordinate 1. 2. It produce simple formulation for handling the lower boundary layer. 3. Allows for good depiction of continuous fields such as temperature advection and wind.(lee mountain slope) 4. The model can better define boundary layer processes such as low-level wind, turbulence etc.

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The disadvantages of the σ coordinate 1. The coordinate surfaces slope steeply to follow steep mountains. 2. Horizontal derivative calculations yield errors in the vicinity of mountains, particularly for pressure gradient force 3. Errors increase as model resolution increases with mountain slopes being better represented. 

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conversions from the pressure to the sigma pressure coordinate intersection of pressure, altitude and σ surface in an x-z plane. From the figure, the change in moist-air mass mixing ratio over distance is

q1 − q3 q 2 − q3  σ 1 − σ 2  q1 − q 2  = +   x 2 − x1 x 2 − x1  x 2 − x1  σ 1 − σ 2  Which is an exact equivalence

 ∂q   ∂q   ∂σ   ∂q    =  +  +   ∂x  p  ∂x  σ  ∂x  p  ∂σ  x

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conversions from the pressure to the sigma pressure coordinate A similar equation is written for the ydirection. Written x and y equations in gradient operator from the generalized for any variable gives the gradient conversion from Cartesianpressure to Cartesian-sigma pressure coordinates as:

∂ ∇ p = ∇ σ + ∇ p (σ ) ∂σ

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ETA model 1. The Eta coordinate was developed to try to reduce errors in the computation of the horizontal pressure gradient force that can occur with the sigma vertical coordinate. 2. The ETA model has 60 levels 3. ETA is a hydrostatic model 4. 

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Eta ( or Step ) Coordinate (η) The eta coordinate is, in fact, another form of the sigma coordinate, but uses mean sea level pressure instead of surface pressure as a bottom reference level.

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Eta coordinate

where · p t is pressure at the model top · p (z=0) is the standard atmosphere (1013 hPa) r · p (z ) is the standard atmosphere pressure at r s the model terrain level zs 

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The differences The difference between the definitions of the sigma and eta coordinate systems allows the bottom atmospheric layer of the model to be represented within each grid box as a flat "step," rather than sloping like sigma in steep terrain

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Advantages of the Eta Coordinate

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Advantages of the Eta Coordinate 1. Eta models do not need to perform the vertical interpolations that are necessary to calculate the PGF (pressure gradient force) in sigma models 2. The simple boundary conditions are kept 3. eta models can often improve forecasts of cold air outbreaks, damming events, and leeside cyclogenesis

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Advantages of the Eta Coordinate

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Disadvantages of the Eta Coordinate 1. Little to no convective precipitation over the mountains 2. Boundary level winds too weak and too smooth 850 mb winds too strong 3. Inversions, especially mountain/valley influences, not properly modeled

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summery

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