Modelling Environmental Processes An illustration

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Modelling Environmental ProcessesAn illustration

• Dr Ian Renfrew
• Environmental Sciences

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Overview

• The aim of this course is to show how
  environmental problems may be solved from the
  initial problem, to mathematical formulation and
  numerical solution.
• The course consists of lectures on numerical
  methods and computing practicals. Both are
  extremely important, i.e. compulsory!
• The computing practicals will be run in Matlab.
• The unit will guide students through the solution
  of a geophysical problem of their own choosing.
• The problem will be discussed and placed into
  context through an essay, and then solved and
  written up in a project report. A taught
  practical is also assessed.

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Background

• For UG ENV 2A21 and 2A22
• For MSc
• Some computer programming (any language)
• Some understanding of calculus, in particular
  differential equations

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Learning outcomes

• Start with a geophysical phenomenon
• Determine the key physical/chemical processes
• Literature review, text books, observations,
  laboratory experiments, etc
• Express the key processes in terms of
  mathematical equations
• Formulate a numerical solution to these equations
• Write a computer program to solve the numerical
  equations
• Test, view and analyse the results discuss their
  significance

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An illustration

• A research-led geophysical problem
• Modelling the flow of cold-air off an ice shelf
  and over a polynya (a persistent area of open
  water within the sea ice)
• The model is documented in detail in
• Renfrew, I. A. and J. C. King, 2000 A simple
  model of the convective internal boundary layer
  and its application to surface heat flux
  estimates within polynyas, Boundary-Layer
  Meteorology, 94, 335-356.
• Model then applied to an area in the Southern
  Weddell Sea, where coastal polynyas are common
• Renfrew, I. A., J. C. King, and T. Markus, 2002
  Coastal polynyas in the southern Weddell Sea
  variability of the surface energy budget, J.
  Geophys. Res. (Oceans), 107 (C6), 3063, doi
  10.1029/2000JC000720.

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Coastal air-sea-ice interaction
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Polynyas and Leads
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Why is this important?

• Atmosphere-Ocean heat exchange around Antarctica
  are key part of the oceans thermohaline
  circulation.
• In winter, most heat exchange is thought to take
  place through polynyas and leads (sea ice acts to
  insulate the ocean)
• Need to quantify this heat exchange

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How to quantify the heat exchange?

• Estimate the surface sensible heat flux, surface
  latent heat flux and the radiative fluxes.
• To use standard bulk formulae for the fluxes we
  need to know near-surface air temperature, wind,
  relative humidity, and the sea surface
  temperature.

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The surface energy budget

• QsQl Qr QpQo Qtot pi Lf F
• where
• Qs sensible heat flux
• Ql latent heat flux
• Qr net radiative flux
• Qp heat flux from precipation
• Qo upward heat flux from the ocean
• and
• pi is the density of ice, Lf the latent heat of
  fusion, and
• F an ice production rate.

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The surface energy budget

• Surface sensible and latent heat fluxes can be
  calculated
• Qs CH ?cp U10 (?SST ?m)
• Ql CE ? cp U10 (qsat qa)
• where
• U10 is the wind speed at 10 m
• ?SST and ?m are the potential temperatures at
  the sea surface and in the atmosphere
• qa is the specific humidity
• qsat is the saturated specific humidity at ?SST
• and CH CE are exchange coefficients.

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What are the key physical processes?
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What are the key physical processes?

• Cold air flowing off a cold ice surface over a
  warm ocean surface
• upstream air is stable
• flux of heat from ocean into boundary-layer
  atmosphere
• this will cause an unstable surface-layer which
  will convectively mix upwards through the
  boundary layer
• after convective mixing the boundary-layer ?
  will be constant with height
• a mixed-layer boundary-layer model seems
  appropriate

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What are the key physical processes?
what about upstream temperature profile? 1st
order importance use climatological
information mixing of heat from above? 2nd order
importance but easily encorporated changes in
surface roughness ice to water? 2nd order
importance changes in wind speed? 2nd order
importance literature was ambiguous development
of clouds? 2nd order importance not simple to
model changes in relative humidity? 3nd order
importance qa mainly determined by temp.
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A Convective Internal Boundary-Layer model
Parameters set from climatology ?? stability
- piecewise linear profile hsl initial CIBL
height ß entrainment ratio

• Variables
• U10 constant with x (c.f. literature review)
• ?SST constant with x (ok over 10s km)
• h(x) CIBL height will increase with distance
• ?m(x) will warm with distance
• qa(x) will increase as ?m increases
• Thus Qs and Ql will change with x

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Literature review

• Garratt, JR, 1992 The atmospheric boundary
  layer, Cambridge University Press, page 154
• Outlines simple mixed-layer models,
• When temperature is constant with x then an
  analytic solution is possible (given certain
  assumptions)
• h Cx1/2, where C is a constant and typically
  C(stability,Um,Qs, entrainment)
• In our situation, with ?m(x) and Qs(x) an
  analytic solution is not possible
• Devised an iterative solution to the numerical
  equations.

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Model equations
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Model equations
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Model equations
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Numerical Solution
The model equation set (9), (10) and (11) are
solved by numerical integration, and an iteration
scheme where 1. Hs(xi) is calculated via (11),
using ?m(xi-1) as a first guess. 2. Equations (9)
and (10) are solved for ?m(xi) and h(xi). 3.
?m(xi) is then used to give a revised estimate of
Hs(xi). Steps 2 and 3 are repeated until h
converges to within a defined criteria (set as
one metre), which usually required only two
iterations. The accuracy of the numerical
integration can be checked by comparing Hs from
the bulk formula and as calculated from Equations
(7) and (8) they typically agreed to within 2 W
m-2. The numerical solution outlined here is
rapid enough for climatological use.
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Matlab code

• I have put a simplified version of the CIBL model
  code on my website http//lgmacweb.env.uea.ac.uk/e
  046/teaching/teaching.htm
• cbl_growth_gm.m main code
• Sets up parameters and input variables
• Grows CIBL for successive values of x
• Simple numerical integration to solve equations
  (9) (10)
• Iteration routine to assure convergence
• Simplified model uses a constant heat flux
  coefficient
• cbl_plot_gm.m plotting code
• Simplified for just model solution, no validation
  data
• thermo_rh.m thermodynamics variable function

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Results for a typical cold air outbreak
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Results for 4 February 1997 off the Ronne Ice
Shelf, Antarctica Input data are from an
automatic weather station on the ice
shelf. Validation data are from radiosondes ()
and ship-borne observations (o).
Visible satellite image of Ronne Ice Shelf and
southern Weddell Sea 4 February 1997
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Results for 4 February 1997 off the Ronne Ice
Shelf, Antarctica Input data are from an
automatic weather station on the ice
shelf. Validation data are from radiosondes ()
and ship-borne observations (o).
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Input data from upstream weather
station. Validation data from instrumented
aircraft. Systematic differences are due to CIBL
model limitations. For example, a previous CIBL
development and the development of clouds with
fetch. Note (o) plot total heating fluxes,
while () plot turbulent heat flux convergence
only (i.e. the heating that we model).
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Relevance to Modelling Env Processes

• My illustration was original research that led
  to a publication, your course projects should not
  be as complicated or as lengthy!

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Relevance to Modelling Env Proceses

• The basic principles should be the same
• Determine your geophysical problem
• Simplify to something tractable
• Devise a mathematical model
• Develop a numerical model
• Examine solutions within parameter space
• Discuss their significance
• The first three should be covered in essay
• The whole project covered by the final report