Reservoir Models and Flux Dynamics in Earth Science
Explore the concept of reservoir models and flux dynamics illustrated by a cascade of lakes in Computational Earth Science. Learn how the reservoir flux depends on various factors and its implications on the system. Dive into differential equations and feedback mechanisms affecting reservoir dynamics.
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2023 EESC W3400 Lec 05: Reservoir models illustrated by cascade of lakes Computational Earth Science Bill Menke, Instructor Emily Glazer, Teaching Assistant TR 2:40 3:55
Announcement Second Assignment due Friday nite Emily our TA is back We believe we have solved the Mac animated mpeg problem (tell us if we re wrong)
today: Reservoir Models
Reservoir flux in flux out amount ? ? in this reservoir flux ??? into this reservoir flux ???? out of this reservoir ? ??? ? = ??? ???? differential equation initial condition ? ? = ?0
Feedback: Reservoir flux depends on amount flux in flux out amount ? ? in this reservoir flux ??? into this reservoir flux ????= ?? out of this reservoir positive feedback: ? ??? ? ?? + positive feedback: ? ??? ? ??
Bank Account flux out non-discretionary expenses ? per month discretionary expenses ? per $ per month flux in salary ? per month amount ? ? in $ in this reservoir interest ? per $ per month ????= ? + ?? ???= ? + ?? initial condition ? ? = ?0 in dollars
flux out Reservoir non-discretionary expenses ? = 500 per month flux in salary ? = 1000 per month amount ? ? in $ in this reservoir discretionary expenses ? = 50 12 100 per $ per month interest ? = per $ per month 1 12 100 initial condition ? ? = ?????= 10000 in dollars
M=1; # one variable balance0 = 10000.0; # initial balance salary = 1000.0; # salary in $/mo nondis = 500.0; # non-discretionary spending in $/mo interest = 1.0/(100.0*12); # interest in fraction of balance per month discetionary = 50.0/(100.0*12.0); # discretionary spending in fraction of balance per month # equation for balance u # d/dt u = salary - nondis + interest*u - dis*u; def myfun( t, u ): f = np.zeros((M,)); f[0] = salary - nondis + interest*u[0] - discetionary*u[0]; return f; # initial conditions u0 = np.zeros((M,)); u0[0] = balance0;
discretionary = 50.0/(100.0*12.0); # discretionary spending in fraction of balance per month
discretionary = 90.0/(100.0*12.0); # discretionary spending in fraction of balance per month
your balance flux out flux in non-discretionary salary amount ?0? in $ in this reservoir vendor s balance interest amount ?1? in $ in this reservoir interest initial conditions ?0? = 10000 ?1? = 0
? ?? ?0 ?1 ? + ??0 ? ??0 ??1+ ??0 =
you lose your discretionary spending ? ?? ?0 ?1 ? + ??0 ? ??0 ??1+ ??0 = vendor gains your discretionary spending
M=2; # two variables balance0 = 10000.0; # initial balance balance1 = 0.0; # initial balance of vendor salary = 1000.0; # salary in $/mo nondis = 500.0; # non-discretionary spending in $/mo interest = 1.0/(100.0*12); # interest in fraction of balance per month discretionary = 90.0/(100.0*12.0); # discretionary spending in fraction of balance per month def myfun( t, u ): f = np.zeros((M,)); f[0] = salary - nondis + interest*u[0] - discetionary*u[0]; f[1] = discretionary*u[0] + interest*u[1]; return f; # initial conditions u0 = np.zeros((M,)); u0[0] = balance0; u0[0] = balance1;
note time delay: takes a while before vendor has more $ than you your balance
lakes on a hillside melting glacier seepage thru ground seepage thru ground seepage thru ground
lakes on a hillside ??? ?0 ?/???? ?0/???? ?1 ????? ?1/???? ?2 ????? ?2/????
differential equation ?0 ?1 ?2 ??? ??0 ??0 ??1 ??1 ??2 ? ?? = ?0 ?1 ?2 0 0 0 initial condition at time ? = 0 =
what leaves lake 0 ?0 ?1 ?2 ??? ??0 ??0 ??1 ??1 ??2 ? ?? = flows into lake 1
write this way instead ??? ????? ?0???? ????? ?1???? ????? ?0???? ?0 ?1 ?2 ? ?? ????? ????? ?1???? ?2???? = so at a later time we can examine the case ? ???? = ????? ?0???? ???? ????? ?0 ????
Groups prepare a 1-slide PPTX with your plot of lake levels and email it to me so I can show it in class. MENKE@LDEO.COLUMBIA.EDU put group # in subject
Group 1 Suppose that the flux in stops at ? = ????= 100. FYI, the python expressions (t<tcut) is 1 for times before tcut and zero thereafter). Discuss how the shapes of the lake levels evolve with time, and contrast the shape of the top and bottom lakes.
Group 2 Suppose that the glacial meltwater flux in into the first lake varies periodically according to 2? ?? with ? in the range 50 thru 500 ???= 0.1 cos2 Discuss how the oscillation is expressed in the lake levels of the sequence of lakes. How many lakes is the oscillation noticeable in and how does it vary with ??
Group 3 Suppose that each lake receives the same constant rain flux ?????= 0.015 and that the glacial flux ??? of the first lake is zero Discuss how this new flux is expressed in the lake levels of the sequence of lakes. Explain physically why the lake levels increase with lake number.
Group 4 Suppose that the groundwater flux out of all the lakes is: ? ???? = ????? ? ???? with ? = 4. Describe qualitatively how flux varies with ?. Discuss how this new flux is expressed in the lake levels of the sequence of lakes. Relate your result to flash floods.
