ERTH 430:
Fluid Dynamics
in Earth Systems
Problem #1:
Key to Comments
Dr. Dave Dempsey
Dept. of Earth & Climate Sci.
SFSU, Fall 2016

  1. General Organization

    1. Number (label) each section of your problem solution (say, with a Roman numeral) and title it (e.g., "Relations Needed"). Your solution should have five sections, as outlined in the guidelines for presenting the problem solution that accompanied the problem statement.

  2. Information Given or Known

    1. There is a major difference between the change or difference in a quantity (call it Q) and the rate of change in that quantity and the gradient of that quantity. A change or difference in Q has dimensions of Q, whereas a rate of change in Q has dimensions of Q over time and a gradient of Q has dimensions of Q over distance. These are all entirely different things, even though the rate of change in Q and the gradient of Q both have a change or difference of Q in their denominator (when we approximate them using a finite difference, at least). The language we use to describe and define quantities needs to reflect accurately on what they actually are and what their dimensions are.

    2. The gradient of a field variable is a vector. The information given in the problem about the variation in pressure with respect to position is a scalar—it's one component of the full gradient. The choice of symbol for the component of the pressure gradient given should reflect that difference.

    3. "Tendency" is another name for the local derivative, as noted in the problem statement. The rate of change in (atmospheric) pressure measured on the ship is a total derivative, not a local derivative (and hence not a pressure tendency), because the ship is moving (through the field of pressure).

    4. We are given in the problem that the pressure is observed on the ship to be falling at a rate of 100 Pa/3 hrs. This is the total derivative of pressure (as observed on the ship), dp/dt, the rate of change of pressure as observed on the ship = −100 Pa/3 hrs. If you define dp/dt to be the rate of decrease in pressure observed on the ship, then you'd have to say that dp/dt = 100 Pa/3 hrs. That is, by defining dp/dt as a rate of decrease rather than a rate of change, then a negative rate of change would be a positive rate of decrease. This is usually confusing, and it happens because of the unconventional (and usually unwise) definition of dp/dt as a rate of decrease rather than a rate of change.

    5. Avoid putting boxes around anything in this section--reserve boxed answers for the symbolic solution and the numerical solution in Section IV ("Solution").

    6. The speed and direction of an object (in this case, the ship) are separate pieces of information, so list them separately. (Together they constitute the velocity of the ship, though in this problem it isn't clear whether or not the complete velocity is given, since we aren't explicitly given what it's vertical component of velocity might be.)

    7. Try to use language that distinguishes as clearly as you can between time derivatives and gradients. "Rate of change of pressure in the ship's direction of motion" could be interpreted to be a (somewhat awkward) way to describe the total derivative of pressure measured on the ship, for example. Try to use "rate of change" to describe time derivatives and "gradient" to describe gradients. For example, "the gradient of pressure in the direction of the ship's motion" would be very clear.

    8. Conversion factors are quantities that are assumed known in the background, so we don't list them as "information given or otherwise known" in our problems. They appear when we use them to convert units, but that's usually it.

    9. If a quantity is given a numerical value in the problem statement, assign that value to the symbol you define for the quantity in this section.

    10. For each quantity given or known (and necessary to solve the problem), define it in English and define a shorthand symbol for it (and refer to it using the shorthand symbol thereafter).

    11. When assigning a symbol and a value to a quantity in Section I, "Info Given or Known", I prefer the format:

      * English meaning or definitionsymbol = value and units

      (The symbol "≡" can be read "is equivalent to" or "is assigned the symbol" or "is defined to mean" or "will henceforth be referred to as". )

      If you want or need to convert the units of a quantity defined at this stage, you can write the statement above in general form like this:

      * English meaning or definitionsymbol = value and units × (unit conversion factor) = value and new units
      For example:

      * Gas constant for dry air ≡ Rd = 287 J/kg/K ×(1 kg/1000 gm) = 0.287 J/gm/K

    12. Don't list info in Section I, "Info Given or Known", that is not needed to solve the problem even if it's "given" in the problem. You should have a need for each item listed; there's no reason to list an item that you don't use. (Of course, the same applies to info that you know but aren't given in the problem—there could be a lot of info that isn't needed to solve the problem and that you therefore wouldn't want to list!)

    13. No information is given about the ship's vertical velocity or about how the pressure tendency varies spatially, so if you need information about these, you'll either have to relate them to something we do know or else make an assumption about it. In this case, it's the latter. If you have to make an assumption to solve the problem and the assumption isn't given to you, you should make it in the "Solution" section.

  3. Information Desired

    1. As part of this section, assign a shorthand symbol to the quantity desired (expressed in English). I prefer the format:

      * English meaning or definitionsymbol (in desired units)
      For example:
      * Temperature at top of layer      ≡ Ttop (in °C)
      * Temperature at bottom of layer ≡ Tbot (in °C)

  4. Relations Needed

    1. For each relation that you list, include a (parenthetical) name of the relation (or a brief description if it lacks an official name). "Relation between total and partial derivatives", for example.

    2. In the relation between total and partial derivatives, be sure to break the full, three-dimensional advection term into horizontal and vertical parts. You have to do this because you aren't given the ship's full velocity necessarily, just the horizontal part of it. You also aren't given the full three dimensional pressure gradient, just one part of it, which is horizontal. You want to set yourself up as best you can to connect information given in the problem to the information desired, using symbols that you've carefully defined in Sections I and II. If you introduce new symbols in Section III that represent, or should represent, quantities already defined in Sections I and II, you're setting yourself up for confusion. As hinted at strongly in the assignment description, you'll need to say something about the ship's vertical component of velocity, which hints that you need to define symbols for the horizontal and vertical components of the ship's velocity separately.

    3. Number (label) each relation that you list under "Relations Needed" and all subsequent, intermediate relations that you derive (if any) on the way to a symbolic solution. (Use Arabic numerals, say.)

    4. Some relations applies to any field variable, but when using it for only one field variable in the problem, write the relation using the symbol for that field variable, not a generic symbol (say, Q) that you then have to define to be, say, pressure (as in this problem).

    5. Unit conversions are not relations—they are simply calculations. Hence, don't list unit conversions in this section, which is intended to make clear the basic physical and purely mathematical relations needed to connect the info you know to the info you want to know.

    6. For relations in your list of "Relations Needed", first put an identifying number (an Arabic numeral), then the equation, then the name or description of the relation (in parentheses), then (following and/or below the relation) definitions of any new symbols introduced in the relation. That is, use the general form:

      (#) symbolic relation     (name or description of relation)
      or, when a new symbol is introduced:
      (#) symbolic relation     (name or description of relation), where symbolmeaning or definition
      For example:
      (1) p = ρRdT     (the ideal gas law), where ρ ≡ density of the gas

  5. Solution

    1. Label each significant step of your solution (say, with a capital letter to distinguish it from relation labels and solution-section labels), and narrate it briefly. For example: "(A) Substitute (2) into (1) and solve for Χ", or "(C) Substitute values and units into (3)", or "(B) Apply (2) to the observer in the hot air balloon" (where (1), (2), and (3) in these examples refer to numbered relations). (You don't have to label each step of a series in which you're simplifying units and exponents, gathering numerical values together, etc.)

    2. Develop a symbolic solution first, and put a box around it. (See guidance provided in the problem statement for what a symbolic solution should look like.)

    3. Since the pressure tendency on the ship and on the island aren't necessarily the same in general, we have to make an assumption that the two are approximately the same. Otherwise we can't solve the problem.

    4. Number (label) each relation that you list under "Relations Needed" and all subsequent, intermediate relations that you derive (if any) on the way to a symbolic solution. (Use Arabic numerals, say.) Place derived relations in the "Solution" section beneath the labeled narration of the step that led to the derived relation (see comment E.3 below), not on the same line.

    5. You should never need to restate a relation in Section IV, "Solution" that you've already stated or derived and numbered (labeled) earlier in the solution or in Section III, "Relations Needed". Just refer to it by its number (label). That's what the number (label) is for!

    6. Derive a symbolic solution for a desired quantity first, and only then substitute values and units for symbols. (These should be distinct, and distinctly labeled, steps.) Developing a symbolic solution for a quantity is what we mean by "solving" an equation. Substituting values and units comes after the solution.

    7. Put a box around your final symbolic solution, and number (label) it so you can refer to it by number (label) in subsequent steps (such as the substitution of numbers and units for symbols).

    8. Don't leave units out of your numerical solution at any step. You can (and should) group units together in a clump after you've substituted a value and its units for each symbol in the symbolic solution, but never leave the units out entirely.

    9. Put a box around your final numerical answer (including the symbol you defined for the quantity whose value you've compute).

    10. The first part of the solution is to develop an entirely symbolic solution (with a box around the final result). This is the most powerful and most important part of the whole problem solution because of its generality—it can be used to calculate solutions to lots of other, similar problems. Once the symbolic solution is developed, substitute values and units for all of the known quantities; group all powers of ten together, all units together, and all remaining numbers together; simplify the units; simplify the powers of ten; and last (and probably least), calculate a numerical solution from the grouped numbers (with final adjustments of powers of ten done as needed). (If units still need to be converted—and I prefer that most units be converted in Section I, "Info Given or Otherwise Known", whenever possible—they can be converted either immediately after the substitution of values and units for symbols or at the end, as the final step.)

    11. When you substitute values and units into a symbolic solution and separate the numbers, powers of ten, and units into three separate groups, keep them on the same line. (They are, after all, collectively the answer and have not meaning independent of each other.) Then you can simplify each one step by step in subsequent lines below the initial one, which each group in its own column.

    12. When you group units together, powers of 10 together, and numerical values together separately (after substituting values and units into the symbolic solution), you should do it for individual terms separately, not for the sum or difference of terms collectively. For example, you would write 10 km/3 hrs × 3 Pa/km + 2 Pa/3 hrs = (10×3)(km/3 hrs × Pa/km) + 2 Pa/3 hrs, not = (10×3 + 2)(km/3 hrs × Pa/km + Pa/km.

  6. Check Solution

    1. Always check your solution.

    2. When checking your solution, units and sign are things you can almost always check. Magnitude is typically trickier, but you can sometimes refer to magnitudes in more familiar situations for comparison, if you know one. In this problem you can argue that in addition to being negative, the pressure tendency has to be greater than the term accounting for the rate of increase pressure that the ship should experience due solely to the fact that it's moving toward higher pressure, because otherwise the pressure would be observed to be falling as measured on the ship. Otherwise, though, you lack context for deciding whether the magnitude is reasonable because ERTH 430 hasn't given you any such context. (It turns out to be reasonable, though a little on the high end of the reasonable range.)

  7. Units and Unit Conversions

    1. There are 100 Pascals in 1 mb, as stated in the problem.

    2. In this problem it turns out that it's better to convert Pascals to millibars at the end rather than in Section I. Doing it at the end means converting units on only one quantity rather than having to do it for two.

    3. Show your unit conversions explicitly—don't simply replace a value expressed in one set of units with another value with another set. E.g., write "p = 10 mb × 100 Pa/mb = 1000 Pa", not "p = 10 mb = 1000 Pa", or simply replace 10 km/hr with 30 km/3 hrs--show the conversion from units of hours to units of 3 hours explicitly.

    4. Unfortunately, we can't add numbers together that have different units attached to them. For example, we can't add 60 seconds + 1 hour and get 61 seconds or 61 hours! Similarly, a unit of 1 hour and a unit of 3 hours are different time units, so we can't say that 1 mb/3 hrs − 91.5 mb/hr = −90.5 mb/3 hrs.

    5. 30 x 3.5 = 101.5, not 91.5.

    6. Convert units (if you choose to) in the same statement that defines the quantity in English and assigns symbol, not as a separate step somewhere.

    7. Whenever you introduce a numerical value into the problem, always attach its units (if it has any).

    8. I've come to prefer that units for most quantities be converted in Section I, "Info Given or Known", as part of the same statement that defines a symbol and assigns a value (see 1.b. above). For a quantity (such as the information desired) the value of which you won't know until you're calculating a numerical solution at the end, convert the units at the end as needed. One exception: don't convert units for multiple quantities in Section I when you can accomplish the same unit conversion just once at the end of Section IV, "Solution".

    9. Don't list unit equivalences (such as "1 mb = 100 Pa") as "information given or known"—that's not the kind of information that we mean to put in this category. Invoke unit conversions as multiplicative factors when converting units; they don't need to appear anywhere else or in any other form.

    10. If you convert a higher-order unit, such as a Joule, Pascal, or Newton, into lower-order units (such as meters, seconds, or kilograms, or Newtons in the case of Joules and Pascals), be clear about the conversion by placing the converted units in your solution expression in the step immediately after the one where the higher order unit last appeared.

    11. The unit conversion term from °C to Kelvins is 273.15K, an additive constant (rather than a multiplicative factor).

    12. Since there are 100 centimeters in one meter, it follows that there are (100 cm) × (100 cm) × (100 cm) ≡ (100 cm)3 = 106 cm3 in 1m × 1m × 1m ≡ 1 m3. That is, the conversion factor from cm3 to m3 is (1 m/100 cm)3 = (10-2 m/cm)3 = 10-6 m3/cm3.

    13. The conversion factor from Pascals (Pa) to millibars (mb) is 1 mb/100 Pa. A Pascal is 1 Newton/m2.

    14. Quantities always retain their basic dimensions, even you change the units used to quantify each of those dimensions. For example, the gas constant for any gas has dimensions of (energy/mass)/temperature. In the MKS system, its units would be (Joules/kg)/Kelvins, while in the CGS system it would be (ergs/gm)/Kelvins. You can convert the units any way you want, but in the end the dimensions must still be (energy/mass)/temperature. You can sometimes catch mistakes that you've made in manipulating symbols to get a symbolic solution or in manipulating (or even simply assigning) units to get a quantitative answer, simply by checking to make sure that the dimensions are correct. (Checking the units in Section V ("Check Solution") at the end, not only makes sure that you've got the units desired for the solution but also effectively checks dimensions, too—you can't have the wrong dimensions and also have the right units.)

  8. Symbols

    1. In this problem we're asked to find the pressure tendency at a nearby island, but the relations available to us can tell us only about the pressure tendency at the ship's location. In principle these aren't necessarily the same, so the symbols used to represent pressure tendency need to reflect the potential difference. (You'll ultimately have to assume that they are equal in value, but that's a different question.) One conventional way to represent the difference is to attach a vertical bar to the tendency symbol with subscript to indicate where the tendency applies: ∂p/∂t|ship vs. ∂p/∂t|island. Another way to do it would be to write p as a function of location and time, and subscript the vector location: ∂p(rship,t)/t vs. ∂p(risland,t)/t.

    2. You can define the total derivative of pressure as observed on the ship simply as dp/dt. The additional information about where and when will be included in this definition without representing it explicitly.

    3. We have a conventional symbol (s) for position along an axis aligned with an object's trajectory, and a conventional symbol for eastward (x) in a rectangular coordinate system. Use one of these in the symbol for the pressure gradient in the direction of the ship's motion or eastward, not "H". (Note that if x increases eastward, then the gradient in that direction is negative in this problem.)

    4. Attaching a subscript "s" to the pressure in the symbol for the pressure gradient in the direction of the ship's motion can be confusing—better not to subscript the pressure at all in this problem since the only pressure involved is atmospheric pressure (a field variable).

    5. Use conventional choices for symbols whenever you can. For example, in our class, a conventional choice for the ship's horizontal speed might be cH. Writing it as "D/T" is highly unconventional and therefore confusing. A conventional choice for the gradient of pressure in the direction of the ship's horizontal velocity would be ∂p/∂s. Writing it as P0/D is highly unconventional and therefore confusing. Same for dp/dt. Writing this as −P2/3T is really confusing. (Symbols represent numerical values, which include a sign—as a general rule, symbols shouldn't include signs or values as part of the name of the symbol. For example, what if dp/dt were positive? What if it's value was represented using mb/hr instead of mb/3 hrs? The same symbol represents all of those numerical possibilities, which is part of the power and generality of using symbols in the first place.)

    6. When defining symbols, strive for consistency. For example, if there are several different temperature values in the problem, use "T" to represent temperature generally and put an appropriate subscript on each T symbol representing a different temperature. If a particular object has several properties, such as density and mass and temperature, and you choose to put a subscript on at least one of the symbols representing these properties to tie them more clearly to the object, then use the same subscript on all of the symbols. (For example, if TH is the temperature of air inside an empty skull, use mH, ρH, and VH to represent the mass, density, and volume of the air in the skull, respectively.) If there is only one object with these properties, you can dispense with a subscript tying it to the object altogether if you want—the meaning of the unsubscripted symbols will be unambiguous when you define them. However, a symbol with a subscript and the same symbol without one are not the same symbol.
    7. Don't change symbols representing the same quantity part way through the solution, and don't introduce new symbols without also defining them (in English).

    8. Once you've defined a symbol, you can use it whenever you need it thereafter and don't have to define it again.

    9. There's no need to define symbols that don't appear in any of your relations.

    10. Try to be efficient when defining symbols so that you don't have to define more than you need. If you define one set of symbols for information given, otherwise known (and needed), and desired, then use the same symbols in the relation(s) that you list in Section III ("Relations Needed"). Otherwise you first have to define the new set of symbols that appear in the relation(s), then state that the first set of symbols are equivalent to the ones in the relations. This is unnecessary work.

      One strategy is to write relations in Section III using generic symbols, as we're used to seeing them written (for example: "p = ρRdT" for the Ideal Gas Law), and if the relation is applied in the problem to only one context, then assign these same symbols to specific information given or desired in the problem. That way, there is no confusion about which values (with units) you should substitute into symbolic solutions, because you will have defined them from the beginning consistently.

      In contrast, if you had defined quantities given in the problem as "Pressure of dry air inside bike tire ≡ ptire = 3.0×105Pa", "Gas constant for dry air ≡ Rd = 287 (J/kg)/K", and "Temperature of air inside bike tire ≡ Ttire = 293K", and you have a symbolic solution for the density of air inside the bike tire of the form "ρ= p/(RdT)", then you can substitute only the known value of Rd into this symbolic solution because T and p are not known. That is, T and Ttire don't represent the same thing and p and ptire don't represent the same thing, because they are different symbols. However, if you had defined T and p to represent the temperature and pressure of dry air inside the bike tire from the beginning, then you can substitute the values for these quantities directly into the symbolic solution because the symbols are the same and are unambiguously defined.

      The situations in which this strategy doesn't work is when you need to apply your relation(s) to more then one context with different sets of values of the quantities involved. In that case, define symbols generically when you state the relation(s), and attach subscripts to symbols defined for quantities given, known, or desired. Then you can (1) derive a generic symbolic solution; (2) "apply" it to each specific context, rewriting the generic symbols using the subscripted versions in the process to get a symbolic solution for the specific context; and (3) substitute values and units for the subscripted variables in the context-specific version of the symbolic solution. You would then repeat this for each distinct context. This removes any ambiguity about what the symbols mean and reduces the chances that you or your audience will get symbols confused and make mistakes.

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