Modelling Environmental Processes

An illustration

Dr Ian Renfrew

Environmental Sciences

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.

Background

• For UG ENV 2A21 and 2A22

• For MSc– Some computer programming (any language)– Some understanding of calculus, in particular

differential equations

Week Lecture: 12-13 Mondays Practical session: 9-12 Thursdays Course work

1 Overview Practical 1 – Matlab tutorial & Project Discussion 1

Essay set

2 Numerical Methods … Practical 2 – Matlab programming & Project Discussion 2

3 … Practical 3 – Vortex motion

4 … Practical 4 – ODEs Essay due

5 … Practical 5 – Diffusion equation Practical set

6 … Talks on project topics – 10 mins each

7 … Practical 6 – Advection equation Practical due

8 last lecture Practical 7 – Boundary-value problems

9 Project – Lab D Project

10 Project – Lab D Project

11 Project – Lab D Project

12 Project – Lab D - Project report due

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

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.

Coastal air-sea-ice interaction

Polynyas and Leads

Why is this important?

• Atmosphere-Ocean heat exchange around Antarctica are key part of the ocean’s 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

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.

The surface energy budget:

Qs+Ql + Qr + Qp+Qo= 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.

The surface energy budget:

Surface sensible and latent heat fluxes can be calculated:

Qs= CH ρcp U10 (θSST – θm)

Ql= CE ρ cp U10 (qsat– qa)

where

U10 is the wind speed at 10 m

θSST and θm are the potential temperatures at the sea surface and in the atmosphere

qa is the specific humidity

qsat is the saturated specific humidity at θSST

and CH & CE are exchange coefficients.

What are the key physical processes?

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 θ will be constant with height • a “mixed-layer” boundary-layer model seems appropriate

What are the key physical processes?

what about:upstream temperature profile?

1st order importance – use climatological informationmixing of heat from above?

2nd order importance – but easily encorporatedchanges in surface roughness – ice to water?

2nd order importance changes in wind speed?

2nd order importance – literature was ambiguousdevelopment of clouds?

2nd order importance – not simple to modelchanges in relative humidity?

3nd order importance – qa mainly determined by temp.

A Convective Internal Boundary-Layer model :

Variables:

U10 ~constant with x (c.f. literature review)

θSST ~constant with x (ok over 10s km)

h(x) CIBL height will increase with distance

θm(x) will warm with distance

qa(x) will increase as θm increases

Thus Qs and Ql will change with x

Parameters set from climatology: γθ stability - piecewise linear profile hsl initial CIBL height β entrainment ratio

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 θm(x) and Qs(x) an analytic solution is not possible

• Devised an iterative solution to the numerical equations.

Model equations

Model equations

Model equations

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 θm(xi-1) as a first guess.

2. Equations (9) and (10) are solved for θm(xi) and h(xi).

3. θm(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.

Matlab code

• I have put a simplified version of the CIBL model code on my website http://lgmacweb.env.uea.ac.uk/e046/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

Results for a typical cold air outbreak

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

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).

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).

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!

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