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Modelling Environmental Processes An 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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WeekLecture: MondaysPractical session: 9-12 ThursdaysCourse work 1OverviewPractical 1 – Matlab tutorial & Project Discussion 1 Essay set 2Numerical Methods …Practical 2 – Matlab programming & Project Discussion 2 3…Practical 3 – Vortex motion 4…Practical 4 – ODEsEssay due 5…Practical 5 – Diffusion equationPractical set 6…Talks on project topics – 10 mins each 7…Practical 6 – Advection equationPractical due 8last lecturePractical 7 – Boundary-value problems 9Project – Lab DProject 10Project – Lab DProject 11Project – Lab DProject 12Project – Lab D-Project report due

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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, 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: /2000JC

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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: Q s +Q l + Q r + Q p +Q o = Q tot = p i L f F where Q s = sensible heat flux Q l = latent heat flux Q r = net radiative flux Q p = heat flux from precipation Q o = upward heat flux from the ocean and p i is the density of ice, L f 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: Q s = C H ρc p U 10 ( θ SST – θ m ) Q l = C E ρ c p U 10 ( q sat – q a ) where U 10 is the wind speed at 10 m θ SST and θ m are the potential temperatures at the sea surface and in the atmosphere q a is the specific humidity q sat is the saturated specific humidity at θ SST and C H & C E are exchange coefficients.

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What are the key physical processes?

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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 θ 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? 1 st order importance – use climatological information mixing of heat from above? 2 nd order importance – but easily encorporated changes in surface roughness – ice to water? 2 nd order importance changes in wind speed? 2 nd order importance – literature was ambiguous development of clouds? 2 nd order importance – not simple to model changes in relative humidity? 3 nd order importance – q a mainly determined by temp.

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A Convective Internal Boundary-Layer model : Variables: U 10 ~constant with x (c.f. literature review) θ SST ~constant with x (ok over 10s km) h(x) CIBL height will increase with distance θ m (x) will warm with distance q a (x) will increase as θ m increases Thus Q s and Q l will change with x Parameters set from climatology: γ θ stability - piecewise linear profile h sl initial CIBL height β entrainment ratio

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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 = Cx 1/2, where C is a constant and typically C(stability,U m,Q s, entrainment) In our situation, with θ m (x) and Q s (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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Numerical Solution The model equation set (9), (10) and (11) are solved by numerical integration, and an iteration scheme where: 1. H s (x i ) is calculated via (11), using θ m (x i-1 ) as a first guess. 2. Equations (9) and (10) are solved for θ m (x i ) and h(x i ). 3. θ m (x i ) is then used to give a revised estimate of H s (x i ). 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 H s 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 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

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