Front matter

Preface

This book grew out of lecture notes for a graduate-level fluid mechanics course that I teach at the University of Miami’s Rosenstiel School of Marine, Atmospheric, and Earth Science. The course provides advanced undergraduate students or beginning graduate students with a solid foundation in fluid mechanics concepts that are essential for understanding ocean physics. While there are many excellent fluid mechanics textbooks available, most are written for either for engineering students with a focus on industrial applications, or for students of meteorology and oceanography that focus on geophysical flows. Many of our students at Rosenstiel go on to pursue research topics related to turbulence, boundary layers, and ocean surface waves, and these topics are well covered by different textbooks in these respective subdisciplines. This book bridges that gap by presenting a concise yet comprehensive treatment of classical, geophysical, turbulent, and wavy fluid mechanics processes that are relevant to atmospheric and ocean physics research.

The book progresses from fundamental concepts to increasingly complex topics. We begin with a review of vector calculus before introducing core fluid mechanics principles such as conservation of mass and momentum. The effects of Earth’s rotation and density stratification, which are crucial for atmospheric and ocean dynamics, come next. We then explore simplified yet powerful models like the shallow water equations which allow an analytical examination of most common solutions. Later chapters cover turbulence, boundary layers, and surface gravity waves - phenomena that are ubiquitous in the ocean and that are becoming increasingly important in coupled weather-ocean prediction and climate projections.

I aim to balance mathematical rigor with physical intuition throughout the text. Detailed derivations are provided, but equal emphasis is placed on understanding the underlying physics. Examples and figures help illustrate key concepts. This textbook is a work in progress and continues to evolve. I welcome feedback from students and colleagues on how to best improve it. I thank my students Stephen Casey, Katia Childs, Katelyn DeWater, Susan Harrison, Jack Lee, Ryland Lewis, Kayla Thompson, Joseph Unsworth, Mia Vallee, and Jessie Yang for their contributions so far. Special thanks to Prof. Mike Brown who previously taught this course and who gave me valuable advice on preparing for it. My hope is that this textbook will serve as a useful resource for students beginning their journey into atmospheric and/or ocean physics research.

Chapter 1

Introduction

What will you learn in this course

The course aims to provide students with a solid understanding of key fluid mechanics concepts that are used in ocean physics research. This course will first refresh you on the vector calculus needed to understand fluid mechanics, and introduce the Eulerian and Lagrangian views of the flow. We then proceed to derive the conservation equations for mass and momentum. We will then consider the effects of rotation and stratification, and look at some steady solutions of the rotating Navier-Stokes equations. Then, we will simplify the flow by looking at it as a thin layer of incompressible rotating fluid that flows over a variable bottom topography and a free surface. From there, we will study turbulence and boundary layers, and complete the course with the linear theory of surface gravity waves. By the end of the course, you will be proficient in applying fluid mechanical concepts and mathematical tools to solve many ocean physics research problems. It will also prepare you for more specialized courses on turbulence, waves, and atmospheric and oceanic circulation.

Reference textbooks

No single textbook out there covers all the topics that we need for this course. However, parts of this course are covered in detail by various textbooks. These lecture notes are based on the following textbooks:

1. Fluid Mechanics, 7$^{th}$ Ed., by Kundu, Cohen, Dowling, and Capecelatro (Elsevier), for the classical (Kundu et al., 2024);

2. Atmospheric and Oceanic Fluid Dynamics (AOFD) by Geoffrey Vallis (Cambridge University Press), for the geophysical (Vallis, 2017); but see also his more concise Essentials of Atmospheric and Oceanic Dynamics (Cambridge University Press), for a shorter and more selective treatment (Vallis, 2019);

3. Turbulent Flows by Stephen Pope (Cambridge University Press), for the turbulent (Pope, 2001);

4. Water Wave Mechanics for Engineers and Scientists by Dean and Dalrymple (World Scientific), for the wavy (Dean and Dalrymple, 1991).

While the notes contain the distilled and required information for you to succeed in this course, please refer to these textbooks for more detailed explanations and examples. Over time, the lecture notes will evolve toward a unified, coherent, and self-contained book.

Chapter 2

Review of vector calculus

In this section we will review the necessary concepts from vector calculus that we will use in this course. These include: scalars, vectors and tensors; gradient, divergence, and curl; line, surface, and volume integrals; and the Gauss and Stokes theorems.

Scalars, vectors, and tensors

In this book we will use three types of quantities to describe fluid properties: scalars, vectors, and tensors.

Scalars are completely described by their magnitude. Examples of scalars are temperature, pressure, or density. A value of 290 K, for example, completely describes the temperature of a fluid at some point in space and time. The fundamental scalar fields in fluid mechanics are the pressure, density, and in some derivations, the velocity potential. In the atmosphere, density is often represented with the air temperature and humidity scalars through the ideal gas law. In the ocean, density if typically represented with the water temperature and salinity scalars through the equation of state. The fundamental scalars for us are then, in this approximate order, pressure, density, temperature, water salinity, and air humidity. In equations, we will write scalars using italics, e.g. $T$, $p$, or $\rho$.

Vectors have a magnitude and a direction. Examples of vectors are velocity, acceleration, or force. In 3-dimensional Cartesian space with coordinates $(x, y, z)$, for example, vector $\mathbf{u}(x,y,z)$ can be described by its components

where $u_x$, $u_y$, and $u_z$ (each a scalar) are the components of $\mathbf{u}$ in the $x$, $y$, and $z$ directions, respectively. This is the general conventional notation, however, we will often write vectors inline as $\mathbf{u} = (u, v, w)$. The fundamental vector field of fluid mechanics is the velocity. Many other vector fields are derived from velocity, such as vorticity, acceleration, and force. In equations, we will write vectors using boldface, e.g. $\mathbf{u}$, $\mathbf{a}$, or $\mathbf{F}$.

The magnitude, or norm, of a vector $\mathbf{u}$ is written as $||\mathbf{u}||$ and calculated as

Here we’re working in 3-dimensional Cartesian space, but vectors can be defined in any number of dimensions, and the above definitions generalize exactly how you’d expect them to. The most ubiquitous vector field in fluid mechanics is the velocity. In atmospheres and oceans, we will often refer to the velocity as wind and current, respectively. Wind speed is thus the magnitude (norm) of the wind vector, and likewise for the current speed.

Tensors have magnitude, direction, and orientation. They are vectors that act on each respective surface orthogonal to the direction of the tensor. Arguably the most important tensor in fluid mechanics is the stress tensor. In 3-dimensional space, for example, a stress tensor can be described as:

In this notation and index ordering, i.e. $\tau_{ij}$, the first index ($i$) refers to the direction of the stress component, and the second index ($j$) refers to the direction of the normal to the surface. In other words, each row of the tensor contains the three components of a vector, and each column contains the three surface normals that the stress component is acting on. For example, $\tau_{xy}$ is the stress in the x-direction and is acting on the surface whose normal is in the y-direction (and which lies in the x-z plane).

One special type of tensor is the identity tensor $\mathbf{I}$, which is a tensor that maps a vector onto itself. In Cartesian coordinates, it is given by:

It may be useful to think of scalars as 0$^{th}$-order tensors, vectors as 1$^{st}$-order tensors, and tensors as 2$^{nd}$-order tensors.

Unit vectors

Unit vectors are vectors with magnitude of 1. A popular notation for unit vectors in Cartesian coordinates is $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$, which point in the $x$, $y$, and $z$ directions, respectively. So, a vector $\mathbf{u}$ can be written as

Notice that you can get the unit vector by dividing any vector by its magnitude, i.e. $\mathbf{u}/||u||$.

Vector operations

Two vectors can be added, subtracted, or multiplied. Although vector addition and subtraction are straightforward (simply add or subtract each of their respective scalar components), vector multiplication is more interesting. There are many ways to multiply two vectors, but the two most important ones for us are the dot product and the cross product.

Dot product

The dot product of two 3-dimensional Cartesian vectors $\mathbf{a}$ and $\mathbf{b}$ is an element-wise sum of their components (and thus, a scalar!):

More generally, the dot product of two n-dimensional vectors $\mathbf{a}$ and $\mathbf{b}$ is

The dot product is commutative, meaning that $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$.

The magnitude of a dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle $\theta$ between them:

To visualize this relationship, take one vector and project it onto the other. This projection is the magnitude of the vector times the cosine of the angle between them. Now, one vector and the projection of the other onto the first vector are pointing in the same direction, so their dot product is the product of their magnitudes. It can be useful to think of a dot product as collapsing the two vectors into a single scalar that contains contributions from each of their components.

The following listing shows how to manually compute the dot product of two vectors in Python using the built-in arithmetic operators:

import numpy as np

# initialize two vectors; specific values are arbitrary.
a = np.array([1, 2, 3])
b = np.array([4, 5, 6])

c = 0  # initialize the result variable
for i in range(a.size):  # loop over indices of the vector
    c += a[i] * b[i]  # multiply elements and add to the result

Notice that this an exact implementation of the right-hand side of Eq. (2.7). The NumPy library, however, allows element-wise multiplication of vectors, which is both more computationally efficient and more concise:

c = np.sum(a * b)  # multiply element-wise and sum up the components

Notice that this is an exact implementation of the middle part of Eq. (2.7). Even though the dot product is simple to implement, as we did above, NumPy provides a function that is even more concise, and likely the most efficient way to compute the dot product:

c = np.dot(a, b)

Although it’s important to understand how to implement the fundamental vector operations by hand, and do it yourself at least once, in practice it’s best to use established libraries such as NumPy, as they are well tested and optimized for computational efficiency.

Cross product

The cross product of two vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as:

where $det(\mathbf{M})$ means the determinant of matrix $\mathbf{M}$.

Using the so-called rule of Sarrus, the cross product can be calculated as:

or:

The result of a cross product is a vector that is orthogonal to both $\mathbf{a}$ and $\mathbf{b}$. Its orientation in space is determined by the right-hand rule: if you point your right thumb in the direction of $\mathbf{a}$ and your index finger in the direction of $\mathbf{b}$, then your middle finger will point in the direction of $\mathbf{a} \times \mathbf{b}$.

The magnitude of the cross product is equal to the product of the magnitudes of the two vectors times the sine of the angle between them:

So, the magnitude of the cross product is largest when the two vectors are orthogonal. Unlike the dot product, the cross product is anticommutative, meaning that $\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}$.

In fluid mechanics, a cross product will often come up when we are interested in the rotation of a vector field. For example, vorticity is the curl of the velocity field.

Matrix multiplication

Occasionally, we will need to multiply a vector by a matrix, or, a matrix by a matrix. As a vector is a special case of a matrix in which either the number of rows or columns is 1, the same rules of matrix multiplication will apply when we multiply a vector by a matrix or a matrix by a matrix. These operations are not commutative, meaning that the order of multiplication matters.

Take two matrices $\mathbf{A}$ and $\mathbf{B}$ such that

and:

The result of their multiplication is a matrix $\mathbf{C}$ given by:

That is, the entry $c_{ij}$ of the product is obtained by multiplying term-by-term the entries of the $i$-th row of $\mathbf{A}$ and the $j$-th column of $\mathbf{B}$, and summing these products. In other words, $c_{ij}$ is the dot product of the $i$-th row of $\mathbf{A}$ and the $j$-th column of $\mathbf{B}$. Although the matrices are not required to be square, the number of columns of $\mathbf{A}$ must be equal to the number of rows of $\mathbf{B}$.

Total and partial derivatives

We will denote total and partial derivative operators (for example, in time $t$) as $\frac{d}{dt}$ and $\frac{\partial}{\partial t}$. Scalars, vectors, and tensors alike can be differentiated with respect to any variable. A derivative of a vector is simply a vector of derivatives of its components:

and likewise for tensors.

Gradient, divergence, and curl

Now, we introduce another operator that builds on top of previous concepts to describe how scalar and vector fields vary in space. This operator is called del and is denoted by the symbol $\nabla$ (pronounced “nabla”):

Written as above, $\nabla$ cannot stand on its own but must be applied as an operator to a field. A good way to think about $\nabla$ is as of a differential operator, which itself is a 3-dimensional vector that can operate on scalars or vectors. Specifically:

• $\nabla p$ is as vector that is a gradient of a scalar field $p$; it quantifies how $p$ changes in space.

• $\nabla \cdot \mathbf{u}$ is a scalar that is the divergence of a vector field $\mathbf{u}$; it quantifies how $\mathbf{u}$ flows out of a point.

• $\nabla \times \mathbf{u}$ is a vector that is the curl of a vector field $\mathbf{u}$; it quantifies how $\mathbf{u}$ rotates around a point.

Although, strictly speaking, one is a symbol and the other is an operator, $\nabla$ (“nabla”) and “del” are often used interchangeably when reading equations out loud.

Gradient

The gradient of a scalar field $T$ is a vector field that points in the direction of the greatest rate of increase of $T$. It is denoted by $\nabla T$ and is defined as

Gradient of a scalar field is a vector that points in the direction of the steepest increase of that field, and its magnitude is the rate of that increase. Imagine hiking up a hill; the gradient of the terrain is a vector that is pointing toward the steepest incline, and its magnitude is the steepness of that incline.

Divergence

The divergence of a vector field $\mathbf{u}$ is a scalar field that describes the rate at which the vector field flows out of a point. It is denoted by $\nabla \cdot \mathbf{u}$ and is defined as

Divergence of a vector field is a scalar that describes how much the vector field is expanding or contracting at a point. Negative divergence is called convergence.

Curl

The curl of a vector field $\mathbf{u}$ is a vector field that describes the rotation of the vector field. It is denoted by $\nabla \times \mathbf{u}$ and is defined as

Curl of a vector field is another vector that is orthogonal to the original vector field and quantifies how much the vector field is rotating around a point. When curl is zero, the vector field is said to be irrotational.

Laplacian

The Laplacian is a second-order differential operator that can be applied to both scalar and vector fields. It measures the rate at which field varies in space and is defined as:

Applied to a scalar field $T$, it is:

Applied to a vector $\mathbf{u} = (u, v, w)$, it is applied to each component:

The Laplacian of a scalar is thus a scalar and the Laplacian of a vector is a vector. In some literature you will see the Laplacian written as $\Delta$, but here we will use $\nabla^2$ to avoid confusion with the $\Delta$ that we use to denote a finite increment.

Useful vector identities

Curl of a gradient of a scalar field is always zero:

Further, divergence of a curl of a vector field is always zero:

Finally, curl of a curl of a vector field is:

Some of these identities will come handy when we derive the conservation of vorticity laws.

Computing and visualizing gradient, divergence, and curl

WIP

Gauss and Stokes theorems

The most useful in our work will be variants of the Gauss and Stokes theorems. The Gauss theorem relates a volume integral of a divergence of a vector field to a surface integral of that vector field. The Stokes theorem relates a surface integral of the curl of a vector field to a line integral of that vector field. Here, they are stated for reference, and we’ll explore their meaning and application in more detail as we use them to derive the fundamental equations for fluid flows.

Gauss theorem

The Gauss theorem states that the volume integral of the divergence of a vector field $\mathbf{u}$ over a volume $V$ is equal to the surface integral of $\mathbf{u}$ over the surface $A$ that encloses $V$:

In other words, the rate of change of the fluid mass within a volume is equal to the flow normal through the surface that encloses that volume. This form of Gauss’s theorem is also known as the divergence theorem. It will come in handy when we derive the conservation of mass (continuity) equation.

Stokes theorem

The Stokes theorem states that the surface integral of the curl of a vector field $\mathbf{u}$ over a surface $A$ is equal to the line integral of $\mathbf{u}$ over the boundary of $A$:

In other words, the rotation rate of the fluid over a surface area is equal to the flow velocity integrated around the boundary of that surface.

Summary

In this chapter, we reviewed:

• Scalars, vectors, and tensors;

• Vector algebra: dot product ($\mathbf{a} \cdot \mathbf{b}$) and cross product ($\mathbf{a} \times \mathbf{b}$);

• Derivatives: total ($\frac{d}{dt}$) and partial ($\frac{\partial}{\partial t}$);

• Gradient, divergence ($\nabla \cdot \mathbf{u}$), and curl ($\nabla \times \mathbf{u}$);

• Gauss theorem that relates volume and surface integrals: $\int_V \nabla \cdot \mathbf{u} dV = \oint_A \mathbf{u} \cdot d\mathbf{A}$;

• Stokes theorem that relates surface and line integrals: $\int_A (\nabla \times \mathbf{u}) \cdot d\mathbf{A} = \oint_{\partial A} \mathbf{u} \cdot d\mathbf{l}$.

These concepts will serve as the basic building blocks for everything that follows in the remainder of this course.

Exercises

1. Pick your favorite programming language (or ask for a recommendation for one). Write a program that defines a scalar, a vector, and a tensor, and assign numerical values to them. Print the values to the screen. Is there a difference in how you define them in your program?

2. What is the dot product of two orthogonal vectors? How about the dot product of a vector with itself? Please write out the solution step by step.

3. Write a program that calculates the cross product of two vectors. Please implement your solution using the basic arithmetic operations such as addition and multiplication. Then, see if your programming language or one of its software libraries provides a function to do this. Can you verify your implementation by comparing its output to that of the library function?

4. How would you calculate a derivative of a quantity (scalar, for example) in a computer program, e.g. $\frac{\partial a}{\partial x}$? Consider that you can approximate a derivative as a difference between two values of the quantity at two points in space. In other words, assume $\partial a \approx \Delta a = a(x_2) - a(x_1)$, and similar for $x$.

5. Write a computer program that calculates the gradient of a scalar field, and the divergence and curl of a vector field.

6. Draw example vector fields that are: (a) non-divergent and irrotational, (b) divergent and irrotational, (c) non-divergent and rotational, and (d) divergent and rotational.

Chapter 3

Fluid kinematics

Fluid kinematics describe the fluid motion without considering the forces that cause that motion. We will explore two main views of the flow: the Lagrangian view, which follows individual fluid particles, and the Eulerian view, which observes the flow at fixed points in space. Although the Eulerian (fixed-point) view is more commonly used in the theory and simulation of fluid flows, the Lagrangian (particle-following) view will be essential when deriving some of the fundamental equations, as well as for understanding where certain features of the flow come from. Both approaches are often used together in numerical simulations. Flows are typically simulated in the Eulerian framework on a fixed grid, and for many applications the flow is analyzed a posteriori and/or visualized in the Lagrangian framework. For example, picture a high-speed flow simulation around an aircraft that is modeled on a fixed grid, and particle-following trajectories drawn to visualize the turbulent wake behind the vessel. Another example is the Lagrangian evolution of an oil spill in the ocean or a volcanic plume in the atmosphere, derived from Eulerian simulation output.

We will also introduce some useful concepts to describe the flow, namely the velocity potential and the stream function. These two scalar quantities are complementary to the vector field of velocity and together provide a complete description of the flow.

Lagrangian and Eulerian derivatives of a fluid property

We will start by first drawing a distinction between the Lagrangian and Eulerian derivatives. Consider a 3-dimensional quantity $\varphi$ that varies in space and time such that $\varphi = \varphi(x, y, z, t)$. This can be a scalar, a vector, or a tensor, however, to keep things simple, suppose $\varphi$ is a scalar field. Let’s find its rate of change. Since it depends on $x$, $y$, $z$, and $t$, the rate of change of $\varphi$ along each of these dimensions must be taken into account. So, the total change of $\varphi$ (let’s call it $\delta\varphi$, where $\delta$ is a small but finite increment) over spatial and temporal increments $\delta x$, $\delta y$, $\delta z$, and $\delta t$, is the sum of changes along each of these dimensions:

Divide by $\delta t$ to obtain:

Recall the definition of $\nabla$ (Eq. 2.18) and let the finite increment $\delta t$ approach $dt$ (and likewise for $\delta x$, $\delta y$, and $\delta z$), to obtain:

The above is equivalent to applying the chain rule to $\varphi$ with respect to time and assuming that the spatial dimension variables are functions of time ($\varphi = \varphi(x(t), y(t), z(t), t)$). Recognize that by stating the dependence of position on time, we are implicitly stating that we are following a fluid particle. Then, recognize that the velocity in each direction is the rate of change of the position in that direction:

which states that the total change of $\varphi$ is due to the local (at fixed point in space) change over time, and due to spatial variations of $\varphi$ as the fluid particle moves through them. Finally, recall the definition of $\nabla$ (Eq. 2.18) to obtain:

The term $\frac{d\varphi}{dt}$ is called the total derivative of $\varphi$. It is also called a Lagrangian derivative, or material derivative, since it follows the motion of a fluid particle. The term $\frac{\partial \varphi}{\partial t}$ is called the Eulerian derivative, or partial derivative of $\varphi$ with respect to time. The term $\mathbf{u} \cdot \nabla \varphi$ describes how $\varphi$ changes due to its spatial variation and the flow of the fluid.

Although the term $\mathbf{u} \cdot \nabla \varphi$ is the dot product of $\mathbf{u}$ and $\nabla \varphi$, the Lagrangian derivative in Eq. 3.5 can be expressed as an operator:

The parentheses on the right-hand side indicate that that term acts as an operator on a field. Like we stated for the operator $\nabla$ in the previous chapter, the total derivative operator $\frac{d}{dt}$ cannot stand on its own, but is instead applied to a field.

Lagrangian derivative of a volume

Consider a fluid parcel with a constant mass but whose volume may change over time and is $\int_V dV = V$. The total rate of change of that volume as it moves with the fluid is equal to the surface integral of the velocity field $\mathbf{u}$ through the surface $S$ that is bounding the volume $V$:

Recall now the divergence theorem (Eq. 2.28) to obtain:

Now, for a volume parcel so small that $\int_V dV = \Delta V \to 0$, the velocity divergence can be considered to be constant over the volume, and the integral can be replaced by the volume itself:

We can derive a similar expression for the rate of change of a fluid property per unit volume $q$, such that $q \Delta V$ is the amount of that quantity in a fluid parcel with the volume $\Delta V$.

Recall the material derivative of $\Delta V$ from Eq. 3.9 to obtain:

This was for a fluid property that is defined per unit volume. Let’s now do the same for some property $\varphi$ that is defined per unit mass, such that $\varphi \rho \Delta V$ is the amount of that quantity in the fluid parcel with the volume $\Delta V$ and density $\rho$ (and mass $\rho \Delta V$).

However recall that our fluid parcel has constant mass, so $\frac{d(\rho \Delta V)}{dt} = 0$. Our total derivative becomes:

The Lagrangian derivative of a volume will come in handy when we derive the continuity equation in the next chapter.

Velocity potential

Velocity potential is defined as a scalar field $\phi$ such that the velocity field $\mathbf{u}$ is the gradient of $\phi$:

The concept of the velocity potential is useful in fluid mechanics because it is often easier to work with a scalar field than a vector field. We will revisit it later in Chapter Surface gravity waves when we derive the equations of surface gravity waves.

Summary

In this chapter, we covered:

• Lagrangian (material) and Eulerian (field) derivatives; the former follows a fluid parcel of constant mass as it moves through the flow field, while the latter is the rate of change at a fixed point (or volume) in space;

• The Lagrangian derivative of volume, as well as of a fluid property per unit volume and per unit mass.

We’ll use these concepts in the next chapter where we derive the equations of continuity and motion.

Chapter 4

Conservation of mass and momentum

In this chapter we will derive the fundamental equations for fluid flows: continuity, momentum, and energy. We start with the conservation of mass, which is the easiest to derive, but also arguably the most fundamental.

Conservation of mass

Recall from the previous chapter that we can take at least two perspectives on the fluid flow: the Lagrangian perspective, which follows a fluid parcel as it moves through space, and the Eulerian perspective, which observes the flow at fixed points in space. We can thus derive the conservation of mass, or commonly known as the continuity, from both perspectives. Let’s start with the Eulerian perspective, as it may seem more intuitive to derive from first principles.

Eulerian derivation

Consider a fixed rectangular volume $\Delta V = \Delta x \Delta y \Delta z$ in three-dimensional space. The mass of the fluid in this volume is $\rho \Delta V$, where $\rho$ is the density of the fluid. The fluid enters the volume through the surfaces of the box, and the rate at which the mass enters the volume through a surface is given by the product of the density, the velocity component normal to the surface, and the area of the surface. Let’s call this velocity $\mathbf{u}$ with components $u$, $v$, and $w$ in the $x$, $y$, and $z$ directions, respectively.

For simplicity, let’s first consider only the $x$-component of the velocity. This scenario is illustrated in Fig. 4.1. The fluid mass flow rate into the volume through the left face is $\rho u \Delta y \Delta z$, and the mass flow rate out of the volume through the right face is $\rho (u + \frac{\partial u}{\partial x} \Delta x) \Delta y \Delta z$. The net mass increase in the control volume must be governed by the net mass inflow excess relative to the net mass outflow:

Now, if we allow the flow field to have components in the $y$ and $z$ directions as well, the equation becomes:

Let $\Delta V \to 0$ to such that any field within $\Delta V$ is uniform to obtain:

This is the continuity equation in the Eulerian reference frame.

We’re not constrained to a rectangular, fixed volume, however. We can derive this equation for an arbitrary control volume using the divergence theorem. The total rate of change of that volume as it moves with the fluid is equal to the surface integral of the velocity field $\mathbf{u}$ through the surface $S$ that is bounding the volume $V$ (Fig. 4.2). Mathematically, we can express this as:

Now, recall the divergence theorem (Eq. 2.28) to obtain:

Let $\Delta V \to 0$ to integrate and drop $\Delta V$ on both sides to obtain Eq. 4.4, which is the Eulerian form of the continuity equation.

Lagrangian derivation

In the Lagrangian frame, we follow a fluid parcel as it moves through space. Its mass $\rho \Delta V$ is constant by definition, but its density or volume may change. Since the mass of the parcel is constant, its Lagrangian derivative is zero:

Since the mass doesn’t change, any change in the density of the parcel must be balanced by a change in its volume:

Recall that we’ve already derived the Lagrangian derivative of a volume of the fluid parcel (Eq. 3.9), which is the second term here. The equation becomes:

Finally, drop $\Delta V$ on both sides to obtain the Lagrangian form of the continuity equation:

Equations 4.4 and 4.10 are two fundamental expressions of the conservation of mass for a fluid. In one form or another, this equation is a critical component of all weather, ocean, and climate prediction models.

Continuity of an incompressible fluid

Liquids are nearly incompressible, and for them $\frac{d\rho}{dt} = 0$ is a good approximation. For an incompressible fluid, the continuity equation simplifies to:

Although as simple as it gets, Eq. 4.11 is extremely important in fluid dynamics.

Conservation of momentum

Like the conservation of mass, the conservation of momentum is a fundamental concept in fluid mechanics. It allows us to predict how the fluid should accelerate due to its state (i.e. velocity and density) and due to the forces acting on it. Together, the continuity and momentum conservation equations form the core of most fluid prediction models, such as weather, ocean, and climate prediction models. We will derive the momentum equation in the remainder of this section. We’ll start from the most basic form first and then incrementally introduce some common forces, such as the pressure gradient force, gravity, and viscosity.

The first step

To derive the momentum conservation equation, we will start from the second Newton’s law, which states that the time rate of change of the momentum of a fluid particle is equal to the net force acting on it. For a fluid parcel of volume $\Delta V = \int_V dV$ whose momentum per unit mass is $\rho \mathbf{u}$, the momentum conservation equation is:

where $\mathbf{F}$ is the net force per unit volume acting on the fluid parcel. Let again the volume parcel be very small such that its density and net force acting on it are uniform. We have:

Recall the Lagrangian derivative operator from Eq. 3.5 to obtain:

This equation states that the acceleration of a fluid parcel at any fixed point in space is equal to the net force per unit mass acting on it, divided by the fluid density. The second term on the left-hand side is the advection term. It represents the local acceleration of the fluid parcel due to the properties of the fluid flow itself. Consider for example a 1-dimensional flow such that the advection term reduces to $u \frac{\partial u}{\partial x}$. Notice that the advection term is zero only in two special cases: when the velocity is zero or when the velocity is spatially uniform. In all other cases the advection term is non-zero and contributes to the local acceleration.

Because the advection term is velocity multiplied by its gradient, it is nonlinear. This single property of this term makes accurate analysis and prediction of fluid flows difficult. For example, the nonlinear advection term is responsible for the existence of chaos in fluid flows, where small differences in initial conditions lead to vastly different outcomes (in popular culture known as the butterfly effect). One consequence of this in our daily lives is that weather predictability is limited to a finite lead time horizon, for example one to two weeks depending on the weather patterns of interest. If, however, we could assume that either the velocity or its gradient are so small that they could be neglected, the equation simplifies significantly and often allows for analytical solutions. $\mathbf{u} \cdot \nabla \mathbf{u}$ is the most important term for turbulence, weather prediction and predictability, and a major obstacle toward analytical solutions of Eq. 4.15 and its variants. Remember this now.

Back to our equation. For a 3-dimensional Cartesian flow where the velocity field is $\mathbf{u} = (u, v, w)$ and net forces are $\mathbf{F} = (F_x, F_y, F_z)$, Eq. 4.15 becomes a system of three equations, one for each component of the velocity field. Recall from the Lagrangian derivative operator that $\mathbf{u} \cdot \nabla \mathbf{u}$ is an operator acting on $\mathbf{u}$ (as opposed to divergence of a gradient). The $\mathbf{u} \cdot \nabla$ operator then expands to $u\frac{\partial}{\partial x} + v\frac{\partial}{\partial y} + w\frac{\partial}{\partial z}$. Our vector equations becomes a system of three scalar equations:

Each of the prognostic equations for the velocity components thus has exactly three advective components that correspond to the gradients of the velocity in each respective direction.

Incorporating the forces

Now we should consider what forces may be acting on the fluid. We distinguish between two types of forces: surface forces and body forces. Surface forces act on the surface of the fluid parcel due to the motion of the fluid molecules, in all directions at that surface. For example, organized motion of molecules into the surface may cause pressure on that surface, and the sheared motion of molecules (e.g. if flow is antiparallel to the surface) may cause shear stress on the surface, leading to the deformation of the fluid parcel. In contrast, body forces act remotely (meaning, from a distance) on the entire volume of the fluid parcel because that parcel is immersed in one or more force fields. Gravity is one such body force, and it’s the only one we’ll consider here. Although in Eq. 4.15 we wrote the net force as $\mathbf{F}$, it’s useful to write it as the sum of body forces $\mathbf{F}_b$ and surface forces $\mathbf{F}_s$:

Let’s derive the surface forces first. We want to find out the local change of momentum only due to the surface forces. Analogous to how the flow through the volume determined the rate of change of density inside that volume, as we saw in the continuity equation (Eq. 4.4), the change in momentum inside the volume is determined by the surface forces acting on the volume (Fig. 4.3).

Mathematically, we can express this change as:

where $\boldsymbol{\sigma}$ is the second-order stress tensor acting on the surface $S$ of the fluid parcel. As before, recall the divergence theorem (Eq. 2.28) to obtain:

The surface force thus equals the divergence of the stress tensor. Insert this into Eq. 4.15 to get our new form of the momentum equation:

This form of the momentum equation is often called the Cauchy momentum equation.

Let’s now look at what this stress tensor divergence term $\nabla \cdot \boldsymbol{\sigma}$ is.

Pressure gradient

There is a fundamental difference in the meaning of the diagonal and off-diagonal components of the stress tensor. The diagonal components of the stress tensor, $\sigma_{xx}$, $\sigma_{yy}$, and $\sigma_{zz}$, represent the normal stress components, i.e. the force per unit area acting on a surface element that is oriented in the $x$, $y$, and $z$ directions, respectively. The off-diagonal components of the stress tensor represent the shear stress components, each acting on all three surfaces. For example, $\sigma_{xy}$ represents the $x$-component of the stress tensor acting on the surface that is perpendicular to the $y$-axis. Let’s write out the stress tensor in Cartesian coordinates:

This tensor can be decomposed into its normal and shear components:

where $p$ is the pressure, $\mathbf{I}$ is the identity tensor, and $\boldsymbol{\tau}$ is the deviatoric stress tensor, or, the viscous shear stress tensor. Written out explicitly in Cartesian coordinates and using Eq. 4.25, the stress tensor is:

The divergence of the stress tensor is then:

Let’s insert this into Eq. 4.15 to get our new form of the momentum equation:

Pressure is one of the fluid properties that determine its state. Collective, organized, motion of molecules at a macroscopic scale induces pressure on a surface and an associated force acting normal to that surface. Recall that the surface vector is normal to the surface and pointing outward, and the force acting on the fluid surface is oriented inward, thus the minus sign.

In an ideal, inviscid fluid, that is, a fluid that exhibits no viscous forces, the stress tensor $\boldsymbol{\sigma}$ is only composed of the diagonal terms (pressure), and the divergence of the stress tensor is zero. Dropping $\nabla \cdot \boldsymbol{\tau}$ and the body forces $\mathbf{F}_b$ for now, the Cauchy momentum equation simplifies to:

This form of the momentum equation is often called the Euler equation.

Viscous forces

Now, let’s look at the shear stress tensor divergence $\nabla \cdot \boldsymbol{\tau}$. Written out explicitly as a matrix of all its components, $\boldsymbol{\tau}$ is:

The diagonal components of the deviatoric stress tensor are the normal stresses, while the off-diagonal components are the shear stresses. The normal stresses are non-zero only in compressible fluids ($\nabla \cdot \mathbf{u} \neq 0$), while the shear stresses are zero in non-viscous flows. The divergence of this tensor, written out explicitly as a matrix of all its components, is:

Now, write out 4.28 as a system of three scalar equations, one for each component of the velocity field, and insert the shear stress divergence terms to get:

Each of the prognostic equations for the velocity components thus has exactly one pressure gradient and two shear stress gradient terms, all arising from the surface forces.

Experimentally, it was found that the viscous shear stress tensor $\boldsymbol{\tau}$ is proportional to the gradient of the velocity field, i.e. $\boldsymbol{\tau} = \mu \nabla \mathbf{u}$. This property of the fluid makes it a so-called Newtonian fluid. The proportionality constant $\mu$ is the dynamic viscosity and depends on the fluid properties and temperature. Inserting this into Eq. 4.28, we get:

We can further simplify this equation by assuming that the viscosity is constant and that the flow is incompressible. This allows us to neglect the viscous stress gradient term, leading to the Navier-Stokes equation.

where $\nu = \frac{\mu}{\rho}$ is the kinematic viscosity. The operator $\nabla^2 = \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right)$ is the Laplacian. It is a second-order differential operator that appears in many partial differential equations, including the heat equation, the wave equation, and the Laplace equation. More on these later.

Let’s now look at the body forces to conclude our derivation.

Gravity

As we mentioned earlier, gravity is the only body force we’ll consider here. The force of gravity per unit mass is given by $\mathbf{g} = (0, 0, -g)$, where $g$ is the gravitational acceleration. Here we assume that the gravitational acceleration is constant and points downward. Insert this into Eq. 4.36, and assuming incompressibility, we get:

Written out explicitly for each of the three spatial dimensions ($x$, $y$, and $z$), we get:

This completes the full system of momentum conservation equations in the Cartesian coordinate system.

Hydrostatic balance

Take Eq. 4.40 and assume that the vertical acceleration $\frac{dw}{dt}$ is small compared to $g$, and that the spatial variations of $w$ are small. We can then drop the $\frac{dw}{dt}$ and $\nu \nabla^2 w$ terms to get the hydrostatic approximation:

which states that the vertical pressure gradient is governed by the density of the fluid and the gravitational acceleration. It’s often a good approximation for large-scale atmospheric and oceanic flows, where the vertical variations of the horizontal velocity components are much smaller than the horizontal variations of the vertical velocity component. Notice however that the hydrostatic approximation does not imply that there is no vertical motion or that it does not vary over time. Instead, according to the continuity equation (Eq. 4.10), it means that the vertical motion is completely governed by the change in density and the divergence of the horizontal velocity field. Further, if the flow is incompressible ($\nabla \cdot \mathbf{u} = 0$), we get:

which relates the vertical acceleration to the horizontal divergence. Integrating this equation vertically allows us to calculate the vertical velocity anywhere in the fluid column provided bottom and top boundary conditions:

This relationship will prove to be extremely useful in ocean applications where only the horizontal velocity field is resolved. For example, a group of ocean surface drifters converging towards a region is indicative of downwelling (downward motion in the ocean) in that region. Another example is that of ocean circulation models, which are typically designed as hydrostatic. In the case of such models, the horizontal components of the velocity are prognostic variables, and the vertical velocity is diagnosed using Eq. 4.42.

Equation of state

Now that we have derived the mass and momentum conservation equations, let’s write them out together in vector form:

Momentum and mass conservation equations are prognostic equations for the vector velocity field $\mathbf{u}$ and the scalar density field $\rho$, respectively. Notice the one remaining unknown: the scalar pressure field $p$. As of now, we have a system of two independent equations for the three unknowns: $\mathbf{u}$, $p$, and $\rho$. We need one more equation to close the system—the equation of state—to relate the pressure to the other properties of the fluid, such as temperature, density, and composition.

The prognostic equations that we have derived so far describe equally well the evolution of both the atmosphere and the ocean, despite their significant differences. The equation of state is where our systems of governing equations for the ocean and the atmosphere begin to diverge. Namely, the atmosphere is a mixture of dry air and water vapor, and the ocean is composed of liquid water with varying amounts of dissolved salts. These differences will reflect in the choice of the equation of state to use in each of these systems.

In the atmosphere

In the atmospheres, ideal gas law is often used as the equation of state:

where $R$ is the specific gas constant for the gas in question, and $T$ is the temperature. For the moist air, we need to account for both the properties of dry air ($R_d \approx\ 287\ J kg^{-1} K^{-1}$) and those of water vapor ($R_v \approx\ 461\ J kg^{-1} K^{-1}$). The equation of state for moist air relies on the so-called virtual temperature to account for the moisture in the air:

where:

where $q$ is the specific humidity of the air. So, the equation of state for moist air is:

Recall that we intended to close our system of equations by finding the equation for pressure. Although we did solve for pressure, it seems that we introduced two new unknown variables: the temperature $T$ and the specific humidity $q$. Each of these variables are governed by their own conservation equations, akin to that for density, but with the addition of source and sink terms that control their production and loss:

where $\dot{S}_T$ and $\dot{S}_q$ are the sources and sinks of temperature and specific humidity, respectively. They are governed by a plethora of thermodynamic processes such as radiation, evaporation, condensation, etc.

Although we won’t delve further into the details behind the sources and sinks of temperature and specific humidity in the atmosphere, we can denote these equations as completing the full system of prognostic equations for the atmosphere: Eqs. 4.44, 4.45, 4.49, 4.50, and 4.51. These equations form the basis of many weather and climate prediction models.

In the ocean

Ideal gas law (Eq. 4.46) doesn’t apply to liquids and the equation of state for seawater is not easily derived. Instead, we assume that the ocean is a single-component fluid, and we use the density field $\rho$ as the equation of state.

where $\rho_0$ is the reference density at the reference temperature $T_0$, salinity $S_0$, and pressure $p_0$. The coefficients $\beta_T$, $\beta_S$, and $\beta_p$ are the thermal expansion coefficient, the saline contraction coefficient, and the pressure coefficient, respectively. This form of the equation of state is a linear equation of state (as in, the dependence of density on temperature, salinity, and pressure each is linear). Dependence of density on temperature and salinity at two different pressure levels is shown in Fig. 4.4. Higher order equations of state are often used for higher accuracy, however they’re out of scope for this course.

Like we did in the case of atmosphere, here we need to additional prognostic equations, one of temperature and another for salinity:

where $\dot{S}_T$ and $\dot{S}_S$ are the sources and sinks of water temperature and salinity, respectively.

Equations 4.44, 4.45, 4.52, 4.53, and 4.54 are the governing equations used in most numerical ocean circulation models.

Nondimensionalization and scaling

A useful technique to simplify the analysis of the governing equations is to scale the variables using characteristic values for each of the variables. This is known as nondimensionalization or scaling the equations. In practice, for each (dependent or independent) variable $x$ in the equations, we define a characteristic value $X$. For example, for the velocity $\mathbf{u}$, we may pick the characteristic value of $U = 1$ m/s or $U = 10$ m/s for the ocean or atmosphere, respectively. We then divide each term in the equations by the characteristic value to obtain a nondimensional (unitless) number. This helps us identify the important parameters that govern the behavior of the system and to group terms in the equations that are of similar magnitudes. This is especially useful for large-scale flows, where the length and time scales can vary over several orders of magnitude.

Let’s look, for example, at the vector equation for horizontal momentum (thus, ignoring $\mathbf{g}$ for now):

The characteristic scales for each term are:

where $U$, $T$, $L$, and $P$ are the characteristic scales for the velocity, time, length, and pressure, respectively. So, if for a given flow we can estimate these characteristic values, we can easily determine which terms are important and which can be neglected. This is the basis of scaling arguments in fluid mechanics.

This approach also enables characterizing the flows in terms of nondimensional numbers. For example, to describe how turbulent or laminar a flow is, it’s useful to relate the inertial to the viscous terms in the momentum equation. Their ratio is called the Reynolds number:

You see that the Reynolds number is proportional to the velocity and length scales each, and inversely proportional to the viscosity. A larger Reynolds number corresponds to a more turbulent flow.

Exercises

1. Derive the Lagrangian form of the continuity equation from the Eulerian form and vice versa. What is the key equation that relates the two forms?

2. Consider two opposing, horizontal, surface currents along the $x$-axis. In the vertical they uniformly span a mixed layer that extends from the surface to the depth of 20 meters, with a magnitude of 1 m s$^{-1}$. The two currents meet at a stagnation zone that is 100 meters wide. Calculate the downwelling velocity at the bottom of the mixed layer. Assume $\nabla \cdot \mathbf{u} = 0$, no change in mean sea level, and no flow in the $y$-direction.

3. Write out the Cauchy, Euler, and Navier-Stokes equations in vector form and discuss their similarities and differences. Give examples of flows that are well described by each of these equations.

4. Write a computer program that calculates the divergence of a second-order tensor in a Cartesian, 3-dimensional coordinate system.

5. Write a function in your favorite programming language that takes a value of temperature, salinity, and pressure and returns the density of seawater. Assume linear dependence of density on temperature, salinity, and pressure. Take the thermal expansion coefficient to be $\beta_T = 1.67 \times 10^{-4} K^{-1}$, the Haline contraction coefficient to be $\beta_S = 7.8 \times 10^{-4} g\ kg^{-1}$, and the compressibility coefficient to be $\beta_p = 4.4 \times 10^{-10} Pa^{-1}$. Take the reference density to be $\rho_0 = 1027\ kg\ m^{-3}$, the reference temperature to be $T_0 = 283\ K$, the reference salinity to be $S_0 = 35 g\ kg^{-1}$, and the reference pressure to be $p_0 = 10^5\ Pa$. When you implement your function, calculate the density of seawater for the range of temperatures from -2 to 30 degrees Celsius, and salinities from 20 to 40 g/kg, and plot it as a contour plot as a function of temperature and salinity. Make such plots for pressure values of $10^5$, $10^6$, and $10^7$ Pa.

6. Calculate the Reynolds number for: (a) a synoptic-scale mid-latitude cyclone in the atmosphere; (b) an mesoscale ocean eddy; (c) a river inflow into the ocean; (d) a breaking ocean surface wave; (e) water flowing through a pipe with a diameter of 0.1 m and flow speed of 1 m s$^{-1}$. Assume $\nu = 10^{-5} m^2 s^{-1}$ for air and $\nu = 10^{-6} m^2 s^{-1}$ for water.