Final Exam Format
The final exam, like our quizzes this semester, will consist of short-answer
questions, mostly about definitions, concepts, relations and the like. You should be able to evaluate (the sign of) terms in the governing equations based on schematic situations, and estimate the magnitude of such terms if provided enough information about the situation. The final will be closed book, closed note, closed computer.
- Reading Questions #1: Continuum Hypothesis, Flow Distortion, and Solving the Governing Equations
- Pre-class Quiz #1: Continuum Hypothesis, Flow Distortion, and Solving the Governing Equations
- Reading Questions #2: Streamlines, Trajectories, and Other Flow Visualization Methods
- In-class exercises and handouts:
- Clicker questions
- Labs (the background and some explanatory text in the instructions for the labs is the most relevant to the final exam):
- Lab #1: The Vertical Pressure-Gradient Force/Mass in the Atmosphere and Ocean
- Part I: Vertical Profiles of Pressure [PDF file]
- Part II: Evaluating Hydrostatic Balance [PDF file]
- Lab #2: Scaling the Terms in the Governing Equations for Waves in a Wave Tank [PDF file]
- Lab #3: Numerical Modeling of 1-D Shallow Water Waves in a Wave Tank [PDF file]
- See Prospectus for Quiz #1 and Prospectus for Quiz #2
- Additional math and physics concepts:
- rough definition (limit of the sum of product of (a) a series of function values and (b) the small increment in an independent variable, over a specified range of values of the independent variable)
interpretation (area under curve on plot of a function vs. an independent variable)
- Fundamental Theorem of Calculus as a relation between
derivatives and integrals.
- the product rule of differential calculus
- the chain rule of differential calculus
- derivatives at local maxima and minima of a function (in particular, on a plot of a function, local maxima and minima have zero slope, so the derivative is zero as such points)
- relation between two quantities, in which changing one quantity by a specified (multiplicative) factor changes the other by the same factor
- constant of proportionality (introduced to convert a proportionality to an equality/equation)
- special case of a linear relation, where the intercept is zero (and where the slope is the constant of proportionality)
(continued from the prospectus for Quiz #1)
- velocity vector components always depend on the type, origin, and axis orientation of whatever coordinate system you choose, because velocity is by definition the rate at which an object's position changes w/r/t time, and position is described in terms of coordinates in a coordinate system:
- In rectangular (Cartesian) coordinates, the scalar components of velocity of a fluid parcel are u ≡ Dxp/Dt, v ≡ Dyp/Dt, and w ≡ Dzp/Dt, where xp, yp, and zp are the coordinates of the fluid parcel's position. These velocity components can be positive or negative, depending on whether the respective coordinate of the parcel is increasing or decreasing (that is, which direction the parcel is moving relative to the orientation of the coordinate axis). Similarly, for a non-fluid parcel, in ERTH 430 we adopt the notation cx≡ Dxobj/Dt, cy ≡ Dyobj/Dt, and cz ≡ Dzobj/Dt, where xobj, yobj, and zobj are the coordinate's of the non-fluid parcel object's position.
- In natural coordinates, the scalar components of a fluid parcel's velocity are horizontal speed ≡ V ≡ Dsp/Dt (which is always positive), horizontal direction, and the vertical component of velocity w ≡ Dzp/Dt, where sp, and an angle that represent horizontal direction, and zp are the three components of the parcel's position. Similarly, for a non-fluid parcel, in ERTH 430 we adopt the notation cH≡ Dsobj/Dt, and cz ≡ Dzobj/Dt, where the three coordinates of the non-fluid parcel object's position are (a) sobj (measured in a horizontal plane), (b) a horizontal angle, and (c) zobj.
- Governing equations
- Conservation of momentum (Newton's 2nd Law)
- three real forces generally act on fluids (and they also apply to solids):
- force of Earth's gravity (a body force)
- the net force due to pressure exerted on the surface of the bit of fluid by surrounding material (a surface force)
- the (net) force of friction (another surface force)
- requires direct contact with another material
- the other material must be moving past the object of interest (or vice versa), resulting in a drag of one object on the other (friction)
- within a fluid, this condition means that there must be shear in the velocity field
- shear is the gradient of velocity in a direction normal to the velocity.
- at the molecular level, friction is analogous in important ways to pressure (except the force of friction acts parallel to an object's surface rather than normal or perpendicular to it)
- in principle friction includes components in all three spatial directions
- mathematical form of the components of (viscous) force/mass of friction in rectangular coordinates:
- (Ffr)x/m = −(μ/ρ)(∂2u/∂y2 + ∂2u/∂z2)
- (Ffr)y/m = −(μ/ρ)(∂2v/∂x2 + ∂2v/∂z2)
- (Ffr)z/m = −(μ/ρ)(∂2w/∂x2 + ∂2w/∂y2)
where μ is the dynamic viscosity (a property of the fluid; it is much larger for liquids than for gases) and μ/ρ is the kinematic viscosity
- the mathematical form of friction/mass tells us that viscous friction depends on the curvature (second spatial derivatives) of the velocity field
- simplified version of Newton's 2nd Law (in the vertical direction): hydrostatic equation
- assume no friction in the vertical direction and no vertical acceleration
- implies balance between (downward) force of gravity and vertical (necessarily upward) pressure gradient force: hydrostatic balance
- hydrostatic approximation is a statement that the material is close to (if not exactly) in hydrostatic balance
- implies that under hydrostatic conditions, pressure must decrease with increasing elevation or altitude in the atmosphere or with decreasing depth in the ocean or beneath the Earth's surface (where it also applies to rock)
- mathematically: 0 = −(1/ρ)∂p/∂z − g
- Conservation of Energy
- Reformulated as conservation of temperature, for most practical applications
- Conservation of Mass
- Reformulated as conservation of density
- The "source" of density for a fluid parcel is the velocity divergence, which describes the fractional rate of change of the parcel's volume as a result of spacial gradients in the velocity (which means that opposite sides of the parcel move at different rates, causing it to stretch or contract, which can cause its volume to change)
- Equation of state
- Ideal Gas Law for gases
- Empirical relationship for liquids (statistical curve fit to careful laboratory observations)
- Because five of the six governing equations describe and explain how physical properties of bits of matter that comprise a physical systems change, they are fundamental for understanding and predicting the behavior of such systems.
- The governing equations allow us to predict the future state, or understand the evolution, of individual parcels of fluid. However, our instruments are fixed relative to the earth (weather stations, stream gauges, anchored buoys, etc.) or sometimes move through fluids attached to airplanes, boats, submersible vehicles, etc., which don't follow fluid parcels (except in rare situations of coincidental good luck). Moreover, we usually want to know how fluid properties will change at fixed locations, not how parcels will change. Hence, to make the governing equations more useful, we want to try to relate material (Lagrangian) derivatives to local (Eulerian) derivatives.
- Relation between total and partial derivatives (can be derived mathematically), applied to any field variable Q:
- dQ/dt = ∂Q/∂t + cx∂Q/∂x + cx∂Q/∂y + cz∂Q/∂x (in rectangular coordinates)
- dQ/dt = ∂Q/∂t + cH∂Q/∂s + cz∂Q/∂x (natural coordinates)
- Relation between material and partial derivatives (special case of the relation between total and partial derivatives)
- DQ/Dt = ∂Q/∂t + u∂Q/∂x + v∂Q/∂y + w∂Q/∂x (in rectangular coordinates)
- DQ/Dt = ∂Q/∂t + V∂Q/∂s + w∂Q/∂x (natural coordinates, where V is the horizontal parcel speed)
- Substitute DQ/Dt from the relation between material and partial derivatives for DQ/Dt in a conservation law for Q and solve for the local derivative (∂Q/∂t) to get a tendency equation for Q:
- ∂Q/∂t = (−u∂Q/∂x − v∂Q/∂y − w∂Q/∂x) + Σ sources and sinks of Q (rectangular coordinates)
- ∂Q/∂t = (−V∂Q/∂s − w∂Q/∂x) + Σ sources and sinks of Q (natural coordinates)
- The tendency of Q is another term for the local or Eulerian derivative of Q (that is, ∂Q/∂t). (The fact that there are three different names for this time derivative must mean that it's important to us!)
- The first term (in parentheses) on the right-hand side of each tendency equation above is the advection of Q. From the perspective looking at a fixed location (not a particular fluid parcel), advection represents a physical mechanism in which a particular fluid parcel leaves the fixed location and is replaced by other, arriving fluid that has its own, possibly different value of Q, which can help account for ("cause") Q to change at the location.
- Our intuition about how rapidly a fluid property might change as a result of advection suggests that advection should depend on:
- the speed of the fluid (the faster it moves, the faster it might bring new values of the physical property to replace the those attached to the departing fluid)
- how different the property is at some distance upstream (or upwind or upcurrent) of the location from the property at the location of interest to start with
- how far away the arriving fluid starts initially
- the latter two items above are captured by the gradient in the property in the direction from which the arriving fluid is coming
- The mathematical form of advection reflects this intuition:
- In rectangular (Cartesian) coordinates:
- advection of physical property Q by the fluid = −u∂Q/∂x −v∂Q/∂y − w∂Q/∂z
- the minus sign is necessary to give the right sense of the change in Q (increase or decrease) at a fixed location occurring as a result of parcels arriving from somewhere nearby with different values of Q than those that departed (you can construct scenarios to demonstrate this yourself)
- In natural coordinates:
- advection of Q = −V∂Q/∂s − w∂Q/∂z (where V is the horizontal speed of the fluid and s is coordinate position along an axis aligned with the horizontal velocity of the fluid)
- the minus sign is necessary because if Q increases with respect to position along the direction of motion of the fluid, then the gradient of Q (in natural coordinates) is positive, but if parcels conserve Q then arriving parcels will have lower values of Q than departing ones (because they are coming from areas of lower Q), which would cause Q at a fixed location to decrease; a similar argument applies if Q decreases with respect to position along the direction of motion (again, the minus sign is necessary to give the right sense of the relation between local changes and the sign of the gradient).
- That is, advection depends on how fast the fluid is moving and on the gradient of Q, which combines the information about how different the fluid property is upstream/upwind/upcurrent and the distance from which it is coming
- Advection isn't the only reason (cause of) why Q might be changing at a fixed location; fluid parcels arriving to replace departing ones might themselves be undergoing changes in Q (that is, they might not be conserving Q), so that they don't have the same value of Q when they arrive as when they started farther upstream or upwind. Mathematically: ∂Q/∂t = (advection of Q) + DQ/Dt
- Replacing DQ/Dt from the conservation law for Q (which gives us the tendency equation) replaces the simple information that Q of a parcel passing through the fixed location might be changing (DQ/Dt), with the causes of that change (the how and why it is changing, in terms of the physical mechanisms by which Q of the parcel is changing).
- Tendency equations look a lot like conservation laws, except that they involve the local derivative (the tendency) on the left-hand side, not the material derivative, and they have an additional source or sink of Q on the right-hand side: namely, the advection of Q.
- That additional source or sink of Q is not a source of sink of Q for a parcel, but rather for fixed locations in space
- Analogous to a conservation law, we can think of both advection and various mechanisms that can cause individual parcels to change their value of Q as "causes" of the local change (or rather, local derivative) of Q.
- The tendency form of the continuity equation
- Following the derivation of tendency equations described above, one form of the continuity equation in tendency form would look (in rectangular coordinates) like this:
- ∂ρ/∂t = (−u∂ρ/∂x − v∂ρ/∂y − w∂ρ/∂z) − ρ(∂u/∂x + ∂v/∂y + ∂w/∂z)
where the first term in parentheses on the right-hand side is density advection and the part of the second term that is in parentheses is the velocity divergence; the minus sign attached to the velocity divergence makes it the velocity convergence; and the result multiplied by density is called the velocity divergence term (a source or sink of density following a fluid parcel) because it is proportional to the velocity divergence
- the velocity divergence, ∂u/∂x + ∂v/∂y + ∂w/∂z, involves gradients of velocity components; it is positive when opposite sides of a parcel move at different speeds in such a way that the volume of the parcel increases; in fact, it can be shown that velocity divergence is just the fractional rate of change of volume of an infinitesimal fluid parcel
- Another derivation is possible, though, starting with a fixed volume of space and considering the rates of flow of mass through each of the sides of the volume. Since the volume is fixed in location and size, any change in the density of fluid inside the volume would occur only because of a change in the amount of mass inside the volume. We can estimate the rate at which mass enters or leaves each face of a cubicle volume (which would depend on the component of velocity normal to each face and the density of fluid entering or leaving the face), sum those rates over all six faces to get the net rate at which mass enters or leaves the volume, and take the limit as the size of the volume and a time increment become infinitesimally small. The result is a version of the continuity equation in tendency form that looks like this:
- ∂ρ/∂t = −[∂(uρ)/∂x + ∂(vρ)/∂y + ∂(wρ)/∂z]
where the term in square brackets is the mass flux divergence (and the minus sign attached to it makes it mass flux convergence).
- A flux is the rate a which some physical quantity passes through, strikes, is reflected from, or is absorbed by a unit of surface area
- u×ρ, v×ρ, and w×ρ have dimensions of (mass/time)/area, so they are mass fluxes
- The gradient of mass fluxes is the mass flux divergence, and minus the mass flux divergence is the mass flux convergence
- If the mass flux convergence is positive, it means that more mass is entering the volume than is leaving it (that is, mass is converging inside the volume), and the density will therefore increase (and of course the local derivative of density will be positive)
- Using the product rule, it is straightforward to show that this flux divergence form of the continuity equation is equivalent to the advective form of the continuity equation above.
- For fluids (such as water) for which the density doesn't respond very much to changes in pressure, we often assume that the fluid is incompressible. That is, following a fluid parcel around, it's density essentially doesn't change (that is, Dρ/Dt = 0). In this case, it follows that Dρ/Dt = ∂ρ/∂t + u∂ρ/∂x + v∂ρ/∂y + w∂ρ/∂z = 0, so that ∂ρ/∂t = −u∂ρ/∂x − v∂ρ/∂y − w∂ρ/∂z. In the advective version of the continuity equation above, this implies that ∂u/∂x + ∂v/∂y + ∂w/∂z = 0, which is an alternative statement that the fluid is incompressible.
- When you squeeze a toothpaste tube, the component of velocity normal to the sides will vary across the width of the tube (for example, if the x- and y-axes are perpendicular to the tube, then
∂u/∂x + ∂v/∂y < 0, which requires that ∂w/∂z > 0. Since toothpaste can't squirt out the sealed bottom of the tube, it follows that w = 0 there. Hence, if ∂w/∂z > 0 and w = 0 at the bottom of the tube, it follows that w > 0 at the top of the tube—that is, toothpaste squirts out the top of the tube when you squeeze the tube. If the cap is on the tube, though, then w = 0 at the top of the tube, too, ∂w/∂z = 0 and so you can't squeeze the tube any narrower (∂u/∂x + ∂v/∂y = 0).
- Simplifying and solving the governing equations
- First note that for any set of coupled equations such as the governing equations, they will have a only one solution (for a given external forcing, such as solar heating, and given set of boundary and initial conditions) if and only if the number of equations and number of dependent variables is the same.
- There are six governing equations (three velocity equations, a continuity equation, a temperature or thermodynamic equation of some sort, and an equation of state) and six dependent variables (typically three velocity components, density, temperature, and pressure)
- If we want to keep track of different constituents of the fluid, such as dissolved salt concentration in seawater, or water vapor concentrations, or liquid and solid water concentrations (in the form of clouds and precipitation in the atmosphere, say), then we introduce a version of the continuity equation for each such constituent instead of one continuity equation for all of them combined
- The governing equations describe, constrain, and predict the behavior of all fluids in all circumstances (on scales significantly larger than the molecular scale, since the equations assume that the continuum hypothesis is true). Given the extraordinary range of fluid behaviors, on scales from microscopic to planetary in size, these equations seem extraordinary. Unfortunately, we don't have a general solution for them.
- To try to make use of the governing equations, we can either simplify them for specific situations or phenomena, or solve them approximately on a computer, or both.
- Simplifying the equations for specific situations or phenomena is typically done using the technique of scaling
- Scaling involves the following:
- identify the phenomenon of interest to you (in our classroom example, it was periodically forced waves in a linear water wave tank, but it could be groundwater flow, or a tornado, or eddies in the surface ocean currents, or disturbances in the jet stream on the scale of the planet, etc.)
- measure typical magnitudes of characteristic features of the phenomena (such as the period, wavelength, and amplitude of waves in our wave tank, the depth of the undisturbed water in the tank, the period and amplitude of the wave generator that provides the periodic forcing of the waves, the density, temperature, and pressure of water in the tank and how much these vary and over what distances and over what time periods)
- Use those measurements to estimate the characteristic magnitude of the terms in the governing equations for the phenomenon of interest; these estimates might comprise rough (often order of magnitude) finite difference estimates of gradients (spatial derivatives) and local derivatives, as well as other factors in the equations (such as the velocity components themselves in the advection terms, or density itself in the continuity equation and in the pressure-gradient force/mass term in the velocity equations)
- neglect all terms in the equations except the largest ones (of similar order of magnitude), and simplify the remaining terms further if we can, also based on scaling
- Once simplified, the equations are easier to understand and interpret and should also be easier to solve
- Can solve them approximately on a computer, typically by:
- defining a set of discrete points in space (often organized in a regular grid)
- approximating the spatial gradients using finite difference approximations based on values at the grid points
- defining a discrete, regularly spaced set of times
- approximating the Eulerian derivative (tendency) using a finite difference approximations based on values at the discrete times
- solving the resulting algebraic (no longer differential) equations for the dependent variables at a series of discrete times
- The approximate (finite difference) equations, together with the specified boundary conditions, and the computer program written to solve the equations, are called a numerical model
- weather forecast models and climate models are examples of numerical models of this sort
- The resulting solutions will not be perfect, even if all terms are kept in them, because the boundary conditions and initial conditions might not be exactly right and because the finite differenced version of the equations is only an approximation of the original differential equations.
- There are many ways to make finite difference estimates of the gradients and local derivatives, and many schemes for solving the resulting algebraic equations, to try to improve the quality of the solutions.
- A whole branch of mathematics (numerical analysis) has developed to do this.
- If the equations are simple enough, it might be possible to integrate them in time and get an exact solution at all locations and times. This is rare, though.
- Part of the solution to the governing equations might include the following:
- Solve one of the equations for one dependent variable, and substitute that solution into the other equations wherever the variable appears.
- This should eliminate one variable from the equations and reduce the number of equations by one
- The equation of state is often used in this way to eliminate density from the equations, for example
- In our periodically forced waves example in class, we found that the hydrostatic approximation was a good approximation
- We solved the hydrostatic equation for the pressure at any arbitrary level, z, in terms of the depth of the water above that level by vertically integrating the hydrostatic equation, and we then substituted for the pressure into the pressure-gradient force/mass term in the horizontal velocity equation, thereby eliminating the pressure from the remaining equations.
- The pressure-gradient force/mass term in the horizontal velocity equation then became proportional to the slope of the water surface (the gradient of the depth of the water).
- For a hydrostatic fluid, this makes sense, for two reasons:
- the pressure at any level in a hydrostatic layer of water depends mostly on the depth of the water above that level (because, if the hydrostatic approximation is valid, then the pressure at any level, z, must balance the force/mass of gravity acting on the column of water above level z—that is, from level z to the water surface)
- the horizontal gradient of pressure at level z would therefore depend on the horizontal gradient of the depth of water above that level, which is just the slope of the height of the water surface
- Integrating the equations with respect to one of the spatial independent variables.
- In our periodically force waves example in class, we vertically integrated the simplified continuity equation over the depth of the water layer, which allowed us to recast the equations in terms of the depth of the fluid (h) instead of the vertical velocity (w).
- This replaced one dependent variable (vertical velocity, w) with another (the depth of the fluid, h), which also appeared in the pressure-gradient force/mass term after we eliminated the pressure.
- This ensured that the number of equations and unknown (dependent) variables remained the same.
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