AOSS 401, Fall 2013 Lecture 4 Material Derivative September 12, 2013 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Cell: 301-526-8572

Class News Ctools site (AOSS 401 001 F13) New Postings to site Syllabus Lectures Homework (and solutions) New Postings to site Homework 1 due today Greene et al. paper: Submit paragraph by Tuesday Francis et al. paper added Discuss on Thursday September 18 Paragraph due on Tuesday September 23

Weather National Weather Service Weather Underground Model forecasts: Weather Underground NCAR Research Applications Program

Outline Questions about homework? Material Derivative Should be review. So we are going fast. You have the power to slow me down.

Newton’s Law of Motion Where i represents the different types of forces.

Our momentum equation Surface Body Apparent This equation is a statement of conservation of momentum. We are more than half-way to forming a set of equations that can be used to describe and predict the motion of the atmosphere! Once we add conservation of mass and energy, we will spend the rest of the course studying what we can learn from these equations.

Our momentum equation Surface Body Apparent Acceleration (change in momentum) Pressure Gradient Force: Initiates Motion Friction/Viscosity: Opposes Motion Gravity: Stratification and buoyancy Coriolis: Modifies Motion

The importance of rotation Non-rotating fluid http://climateknowledge.org/AOSS_401_Animations_figures/AOSS401_nonrot_MIT.mpg Rotating fluid http://climateknowledge.org/AOSS_401_Animations_figures/AOSS401_rotating_MIT.mpg

Consider some parameter, like temperature, T Total variations Consider some parameter, like temperature, T Δx x y Δy (x0,y0)

Taylor series expansion It is sometimes convenient to estimate the value of a continuous function f(x) about a point x = x0 with a power series of the form: In the last approximation, we neglected the higher order terms

If we move a parcel in time Δt Using Taylor series expansion Higher Order Terms Assume increments over Δt are small, and ignore Higher Order Terms

Assume increments over Δt are small Total derivative Total differential/derivative of the temperature T, T depends on t, x, y, z Assume increments over Δt are small

Total Derivative Divide by Δt Take limit for small Δt

Total Derivative Introduction of convention of d( )/dt ≡ D( )/Dt This is done for clarity. By definition: u,v,w: these are the velocities

Definition of the Total Derivative The total derivative is also called material derivative. describes a ‘Lagrangian viewpoint’ describes an ‘Eulerian viewpoint’

In class problem In the absence of pressure forces and viscosity the equations of motion for purely horizontal motion are: Assume that f, the Coriolis parameter, is constant. Calculate the velocity of an atmospheric parcel as a function of position (x,y).

Vector Momentum Equation (Conservation of Momentum)

Vector Momentum Equation (Conservation of Momentum) Coordinate system is defined as tangent to the Earth’s surface x i y j z k east north Local vertical Velocity (u) = (ui + vj + wk)

Tangential coordinates

Previously: Conservation of Momentum Now we are going to think about fluids. Consider a fluid parcel moving along some trajectory.

Consider a fluid parcel moving along some trajectory (What is the primary force for moving the parcel around?)

Consider several trajectories

How would we quantify this?

Use a position vector that changes in time Parcel position is a function of its starting point. The history of the parcel is known

Lagrangian Point of View This parcel-trajectory point of view, which follows a parcel, is known as the Lagrangian point of view. Benefits: Useful for developing theory Very powerful for visualizing fluid motion The history of each fluid parcel is known Problems: Need to keep track of a coordinate system for each parcel How do you account for interactions between different parcels? What can we say about the fluid where there are no parcels? What can we say about the fluid as a whole if all of the parcels bunch together?

Lagrangian point of view: Eruption of Mount Pinatubo Trajectories trace the motion of individual fluid parcels over a finite time interval Volcanic eruption in 1991 injected particles into the tropical stratosphere (at 15.13 N, 120.35 E) The particles got transported by the atmospheric flow, we can follow their trajectories Mt. Pinatubo, NASA animation Colors in animation reflect the atmospheric height of the particles. Red is high, blue closer to the surface. This is a Lagrangian view of transport processes.

Lagrangian Movie: Mt. Pinatubo, 1992 Link to movie on server …. Clicking figure fails on Apple!

Consider a fluid parcel moving along some trajectory Could sit in one place and watch parcels go by.

How would we quantify this? In this case: Our coordinate system does not change We keep track of information about the atmosphere at a number of (usually regularly spaced) points that are fixed relative to the Earth’s surface

Eulerian Point of View This point of view, where an observer sits at a point and watches the fluid go by, is known as the Eulerian point of view. Benefits: Useful for developing theory Requires considering only one coordinate system for all parcels Easy to represent interactions of parcels through surface forces Looks at the fluid as a field. A value for each point in the field – no gaps or bundles of “information.” Problems More difficult to keep track of parcel history—not as useful for applications such as pollutant dispersion (or clouds?)…

Zonally averaged circulation Zonal-mean annual-mean zonal wind Pressure (hPa)

An Eulerian Map

Consider some parameter, like temperature, T x y Material derivative, T change following the parcel

Consider some parameter, like temperature, T x y Local T change at a fixed point

Consider some parameter, like temperature, T x y Advection

Temperature advection term

Consider some parameter, like temperature, T x y

Temperature advection term

Advection of cold or warm air Temperature advection: Imagine the isotherms are oriented in the E-W direction Draw the horizontal temperature gradient vector! pure west wind u > 0, v=0, w=0: Is there temperature advection? If yes, is it cold or warm air advection? warm u y X cold

Advection of cold or warm air Temperature advection: Imagine the isotherms are oriented in the E-W direction Draw the gradient of the temperature (vector)! pure south wind v > 0, u=0, w=0: Is there temperature advection? If yes, is it cold or warm air advection? cold v y warm X

Advection of cold or warm air Temperature advection: Imagine the isotherms are oriented as Draw the horizontal temperature gradient! pure west wind u > 0, v=0, w=0: Is there temperature advection? If yes, is it cold or warm air advection? cold u warm y X

Summary: Local Changes & Material Derivative Local change at a fixed location Advection term Total change along a trajectory

Summary: For 2D horizontal flows with horizontal wind vector and horizontal gradient operator

Conservation and Steady-State

Why Consider Two Frames of Reference? Goal: understanding. Will allow us to derive simpler forms of the governing equations Basic principles still hold: the fundamental laws of conservation Momentum Mass Energy are true no matter which reference frame we use

Movies Eulerian vs. Lagrangian Link to movie on server …. Clicking figure fails on Apple!

Why Lagrangian? Lagrangian reference frame leads to the material (total, substantive) derivative Useful for understanding atmospheric motion and for deriving mass continuity…

On to the Material Derivative…

Material Derivative y Δy x Δx Consider a parcel with some property of the atmosphere, like temperature (T), that moves some distance in time Δt x y Δy Δx

Material Derivative (Lagrangian) Material derivative, T change following the parcel

Local Time Derivative (Eulerian) T change at a fixed point

Change Due to Advection COLD WARM

A Closer Look at Advection Expanding advection into its components, we have

Change Due to Advection

Class Exercise: Gradients and Advection The temperature at a point 50 km north of a station is three degrees C cooler than at the station. If the wind is blowing from the north at 50 km h-1 and the air is being heated by radiation at the rate of 1 degree C h-1, what is the local temperature change at the station? Hints: You should not need a calculator Use the definition of the material derivative and of advection

Class Exercise: Gradients and Advection Material Derivative of T Write in terms of local derivative No east-west or vertical velocity Plug in values

Class Exercise: Gradients and Advection Material Derivative of T Write in terms of local derivative No east-west or vertical velocity Plug in values

Class Exercise: Gradients and Advection Material Derivative of T Write in terms of local derivative No east-west or vertical velocity Plug in values

Class Exercise: Gradients and Advection Material Derivative of T Write in terms of local derivative No east-west or vertical velocity Plug in values

Written in terms of the local change leads to advection Material Derivative Written in terms of the local change leads to advection We will use this again later…

Material Derivative Remember, by definition: and the material derivative becomes Lagrangian Eulerian

Return to the Momentum Equation Remember, we derived from force balances This is in the Lagrangian reference frame In the Eulerian reference frame, we have Non-linear advection of wind velocity This comes from the Eulerian point of view

That’s all