Homework 1
Homework 1¶
How do you write the following phrases using mathematical symbols?
The meridional component of the salinity gradient
The rate of change of temperature (\(T\)) following a fluid parcel
Divergence of fluid velocity
Conservation of mass for an incompressible fluid
Study the geography of winds, gyres, currents, and the terminology used to describe them. For example,
How does the zonal direction of the wind change with latitude? What about the meridional direction across subtropical gyres? Where are mean winds strongest over the North Pacific?
Learn the names characteristics of a few western boundary currents, eastern boundary currents, and equatorial ocean currents. What are the general, qualitative differences between these types of currents?
How do patterns of wind and atmospheric pressure correspond to the locations of subtropical and subpolar gyres?
Learn how to use cyclonic and anticyclonic when describing circulations in the ocean and atmosphere.
What is special about ocean circulation in the Southern Ocean?
In class we developed a model of the greenhouse effect that treated the entire atmosphere as a single layer that passed all shortwave radiation, but absorbed a fraction \(e\) of the outgoing longwave radiation coming from the Earth. Construct a slightly more complicated model by using two layers for the atmosphere. Each layer still passes all shortwave radiation, and each layer absorbs a fraction \(e\) of the incident longwave radiation.
Sketch the model, showing all flux terms. Define \(I\) as the incoming shortwave radiation, \(U\) as the outgoing longwave radiation from the ground, and \(eB_1\) and \(eB_2\) as the longwave fluxes from the lower and upper atmospheric layers, respectively.
Require that the fluxes balance at three levels: above the top layer, between layer 1 and layer 2, and between the ground and layer 1. Write 3 equations in the 3 unknowns, \(U\), \(B_1\), \(B_2\), taking \(I\) and \(e\) as known. Arrange and write these equations as a system of equations and solve for \(U\) as a function of \(I\) and \(e\).
Use the same values for \(I\) and \(e\) that we used in class and calculate the temperature of Earth’s surface. How does this compare with what we got in class? What value of \(e\) would you have to use instead to get a value of \(14^\circ\mathrm{C}\) for the surface temperature?
Check out the most recent measurements of \(\mathrm{CO}_2\) concentration from the Mauna Loa Observatory here in Hawaii: https://www.esrl.noaa.gov/gmd/ccgg/trends/. The increasing atmospheric \(\mathrm{CO}_2\) leads to increasing radiative heating, \(\Delta H\), which varies logarithmically with the increase in concentration for atmospheric \(\mathrm{CO}_2\) (as the effect of increasing \(\mathrm{CO}_2\) on the absorption and emission of long-wave radiation gradually saturates),
\[ \Delta H = \alpha_r\mathrm{ln}\left(\frac{X_{\mathrm{CO}_2}(t)}{X_{\mathrm{CO}_2}(t_0)}\right), \]where \(\alpha_r = 5.4~\mathrm{W\,m}^{-2}\) is a constant that depends on the chemical composition of the atmosphere and \(X_{\mathrm{CO}_2}(t_0)\) and \(X_{\mathrm{CO}_2}(t)\) are the concentrations in parts per million (ppm) for \(\mathrm{CO}_2\) at times \(t_0\) and \(t\).
Using the Mauna Loa \(\mathrm{CO}_2\) data at the link above, estimate the increase in implied radiative heating, \(\Delta H\), in \(\mathrm{W\,m}^{-2}\) between 1958 and now.
Now you have the extra heating at present relative to 1958, use the average extra heating since 1958 (think about how the average relates to the endpoints) to estimate how much the upper ocean might have warmed since 1958. Assume the upper ocean to be \(500~\mathrm{m}\) deep, the density of seawater to be \(\rho=1000~\mathrm{kg\,m}^{-3}\), and the heat capacity for seawater to be \(c_p=4000~\mathrm{J\,kg}^{-1}\mathrm{\,K}^{-1}\).
Loss of heat to the atmosphere due to latent heat flux can have an average value around \(175~\mathrm{W\,m}^{-2}\) in some areas of the ocean. In such a region, What mass of water is evaporated per square meter during one year? Approximately how thick is the equivalent layer of water in meters?
Suppose that the wind speed increases at some point over the ocean. What are the immediate effects on wind stress, sensible heat flux, evaporation, latent heat flux, and buoyancy flux?
Suppose that the sea surface temperature increases at some point over the ocean. What are the immediate effects on sensible heat flux, evaporation, latent heat flux, and buoyancy flux?
A typical value for the tradewinds around Hawaii is about \(15~\mathrm{knots}\), which is about \(7~\mathrm{m\,s}^{-1}\).
Use the parameterization in your notes to compute the magnitude of the wind stress.
Noting that a stress is a force per unit area, compute the mass of water that would have an equivalent downward gravitational force over a \(1~\mathrm{m}^2\) surface area.
What is the volume of this mass of seawater? Use seawater density of \(1000~\mathrm{kg\,m}^{-3}\).
If this volume were spread over the \(1~\mathrm{m}^2\) surface area, how deep would the water be?
How strong would you conclude that wind stress is compared to gravitational forces?
Answer the following questions regarding molecular and eddy diffusion.
Why do we tend to ignore molecular diffusion in favor of eddy diffusion?
Why are eddy diffusion coefficients the same for momentum, heat and salt, but molecular diffusion coefficients differ greatly depending on what is being diffused?
Why are vertical and horizontal eddy coefficients different, while molecular diffusion coefficients are the same in all directions?