28 Atmosphere, Ocean and Climate Dynamics

Vinu Valsala

1. What do you learn from this module?
2. Introduction
1.1 Atmosphere
1.2 Ocean
1.3 Climate
3. Fundamentals of dynamics
3.1 Geophysical fluid dynamics
3.2 Conservation of mass and momentum
3.3 Introduction to potential vorticity
4. Dynamics of the atmosphere and ocean
4.1 Geostrophic and quasi-geostrophic dynamics
4.2 Planetary waves dynamics
4.3 Thermal winds
4.4 Ekman transport in the ocean
4.5 Gyre circulations
5. Climate
5.1 Ocean-atmospheric interaction and climate dynamics
6. Summary

1. What do you learn from this module?

Atmosphere and Ocean are the two most important components of the earth system that are highly mobile. The fluid properties (air and water) of these two entities set up spectacular examples of dynamics that we are (generally) unable to reproduce in a laboratory within a confined area. The most important factor that distinguishes the atmosphere and ocean compared to ‘a fluid in a flask’ is the vast amount of their size and extent. Moreover, the spherical shape of the earth and its rotatory frame of reference (one complete rotation per day) adds more complexities in the dynamics of atmosphere and ocean. In addition to that the differential heating experiencing from the sun (strong heating in the tropics, mild heating in the mid-latitude and very less heating in the polar regions) sets up thermal gradients in the atmosphere and ocean and causes various forms of dynamics.

Climate, on the other hand, is a name we associate to the long term mean behaviour of the atmosphere and ocean that we see around. The interaction of the atmosphere and the ocean (along with the land-biosphere and solid earth) and exchange of matter and properties between them make the long term state of our earth system which we call as climate systems. For this reason, the dynamics of the atmosphere and ocean become the integral part of the climate evolution.

This module will lead you through some of these interesting aspects of dynamic of atmosphere and ocean and will enable you to comprehend and answer some of the features of atmospheric and oceanic circulation that you usually confront in your day to day life. The module will describe about following key points:

1. Physical properties of the atmosphere and ocean
2. Various forcing experiences over the atmosphere and ocean
3. The dynamics of atmosphere and ocean
4. Ocean and Atmosphere as a part of climate system
5. Brief history on climate change

2. Introduction

2. 1 Atmosphere

Atmosphere is the envelope of air surrounding the earth and held closely to the earth surface by the gravitational forces. The atmosphere extends vertically from the surface of the earth upto about a maximum of 100 The density of air at sea level is nearly 1.225 kg m-3 (at 15o C) and decreases rapidly with height and even at 100 miles it is less than that of the best vacuum available in the laboratory.

Thermally the atmosphere has been divided into four layers and they are (1) troposphere (0-10 km), stratosphere (10-50 km), mesosphere (50-80 km) and thermosphere (80 km and aloft). The troposphere is the lowest part of the atmosphere with near-linear decrease of temperature with height (from 273 K to 220 K). The end of troposphere is called ‘tropopause’ which is usually at a 10 km above sea-level where the aeroplanes cruise. The linear decrease of temperature with height in the troposphere is due to the radiative cooling of the atmosphere from surface to aloft.

In the stratosphere the temperature increases with height (200 K to 300 K). This is due to the presence of ozone (O3 molecules) in this layer which absorbs the down-welling ultraviolet radiations from the sun and heats the air there. The cooler tropopause and warmer stratosphere makes this part of the atmosphere very stable (with warm stratospheric air over cold troposphere air) and so the name ‘stratosphere’. The upper part of stratosphere is called ‘stratopause’.

The height above stratosphere is called mesosphere where temperature again falls with height as it cools by radiation. The height between tropopause and 100 km often called as middle atmosphere.

The vertical profile of atmospheric temperature (from surface to 70 km) is approximated as

p = pae(-gz/RTc) = pa e(-z/Hs)……………………………..(1)

where, Hs = scale height defined as RTc/g (Gill, A. E, 1980, page-49). R is the universal gas constant, Tc=250 K and g is the gravity of the earth.

Atmosphere is forced by the differential solar radiation with nearly 300 Wm-2 in the tropics (within ± 230 north-south with respect to the equator) and nearly 50 Wm-2 in the polar region. The outgoing radiation (combination of sensible heat, latent heat and long wave) is nearly 250 Wm-2 in the tropics. The result is a net heating at the tropical atmosphere and cooling at higher latitude. This gradient heating from the sun is the major driver for most of the large scale dynamic and general circulation in the atmosphere.

2.2     Ocean

The 70% of our earth surface is covered by the ocean. The average depth of the ocean is nearly 4000 m. Ocean is the largest storage of water in the earth system. The atmosphere holds only 0.03 m of water if it all precipitates uniformly over the entire global ocean. The Antarctic Ice Sheets is the second largest storage of water, if it all melts, it will add 76 m of water over the entire global ocean. The river and other forms of water storage add up to just 4 m of water over the ocean. Thus oceans, by all means, are the largest reservoir of water in the earth system.

The ocean temperature generally decreases with depth. The first few meters of the surface ocean (nearly upper 50-100 m) are well mixed and properties such as temperature, saltiness or salinity and other nutrients are quiet uniform over this depth. This is called the ‘mixed layer’ of the ocean. Below the mixed layer and upto nearly 200 to 500m the temperature decreases sharply. The most part of the heat stored in the ocean are above this depth. The region of sharp gradient of temperature with depth is called ‘thermocline’ of the ocean.

The composition of ocean water is nearly constant with ion-mass ratio of chloride (55%), sodium (30%), sulphate (8%), magnesium (4%), potassium (1%) and calcium (1%) so that the state of the ocean water can be described by just one major concentration called ‘salinity’. It is usually expressed as the amount of salt in grams per kilogram of sea water. The average salinity of global ocean is 35 0/00 (parts per thousand).

The ocean is heated from the surface unlike the atmosphere which is heated from below. The surface heating makes the ocean temperature larger at the surface and decreasing gradually with depth. This is a stable condition (warm surface water residing over the cold subsurface water). Therefore enormous amount of mechanical energy and thermal cooling is required to mix or stir-up the surface ocean.

Just like the atmosphere, the ocean is also affected by the differential heating from the sun. The tropical oceans are generally warmer and more stratified in vertical than the mid and high latitude ocean. The ocean gains heat in the tropics and transport them to polar region where the heat mediates to the atmosphere. This large scale air-sea exchange of heat is, therefore, tend to level the thermal gradient in the atmosphere and ocean which results in large scale general circulations.

2.3 Climate

It is the long term average of day-to-day state of the atmosphere and ocean or any such element of our earth system. Since the formation of the earth (believed to be some 4.6 billion years ago) the climate is under gradual modification. Due to the reduced radiation pressure in the sun’s core and reduced nuclear fusion, the sun was 30% less luminous during the early evolution of the earth than what it is today. It is a puzzling question what kept the earth atmosphere warm with a 30% fainted sun in the beginning. It is believed that the amount of greenhouse gases in the early atmosphere must have been extremely high (for example CO2 concentration was nearly 20,000 times higher than that of today!). This helped the atmosphere to remain warm and support the early form of microbial life even at 3.0 billion years ago.

The early atmosphere was highly composed of ‘reduced volcanic rocks’. The oxygen was extremely low at that time. It is believed that the long term carbonate-silicate geochemical weathering of rocks, increased photosynthesis and gradual development of a brighter sun all have played intermittent roles to bring the climate amenable for human life as we see today (Kump et al., 2010). Modern climate has an average atmospheric surface air temperature of 15oC. In addition because of the tilt in the earth’s axis of rotation the solar radiation arrives at southern and northern hemisphere is asymmetric and that causes seasons.

3. Fundamentals of dynamics

We have a brief overview of the fundamentals of atmosphere, ocean and our climate from the previous section. In this section, we deal with the dynamics or motion of atmosphere, ocean and climate systems over a definite amount of time.

The fundamental laws governing the dynamics of atmosphere and ocean follow closely the Newtonian classical mechanics. Simply, the differential heating of the atmosphere causes gradient in the horizontal thermal structure which in turn causes a pressure gradient because the temperature, pressure and density of the fluid are thermodynamically connected. This pressure gradient drives the air and water down the pressure gradient. The heating of the atmosphere from the bottom causes air to be less stable and forces convection. The large reserve of potential energy by the gravitational force and thermodynamics (internal energy) could be released by chaotic interaction between the perturbations and the mean-flow. All these cause the dynamics in the atmosphere and ocean.

3.1  Geophysical fluid dynamics

To understand this section one should ask a question ‘what is the difference between the dynamics (motion) of an ordinary fluid in a laboratory and fluid that we see in the atmosphere and ocean?’ Table 1 summarizes the differences between them.

Table 1: Difference between the dynamics of laboratory fluids and planetary fluids

Because of these differences, exciting and unfamiliar (to laboratory fluids) features of dynamics set up in the atmosphere and ocean that we generally call as ‘geophysical fluid dynamics’. For a thorough and detailed discussion on the complexities of geophysical fluid dynamics the readers are encouraged to read Pedlosky, (1987), and Gill A. E., (1980).

3.2     Conservation of mass and momentum

Conservation of mass follows the basic Newtonian principle that the total mass of the fluid remains constant unless it is supplied through external source or consumed/transformed internally and changes its form. Consider a fluid volume of shape of a cube. The total change of mass inside the box is equal to the net rate of mass entering or leaving the box through its all sides. The net balance of the fluid mass can be written as

where, ρ is the density of the fluid and U is the velocity. With sufficient approximation the density can be assumed to be constant over a long time. Moreover the fluid can be assumed as incompressible. This approximation is good with the ocean. In the case of atmosphere considering the decrease in density with height and assuming the atmosphere is dry the quantity ρT is rather conserved, where T is the temperature. With thisassumption the conservation mass represents the equation of continuity which is  whose expansion in three dimensional Cartesian coordinate translates into, where u, v, w are velocities in x, y, z directions, respectively.

Conservation of momentum assumes the Newton’s second law of motion for the fluid. The net forces experiencing on a hypothetical fluid volume will result in the net acceleration of that fluid volume. In the case of ‘laboratory fluids’ this is represented by the famous Navier-Stoke’s equation. And in the case of planetary  fluids, a few additional forces come into picture by virtue of rotating frame of reference and curvature of the surface of the earth.

One such force is the Coriolis force. It is a fictitious force experiencing over a moving body in a rotational frame of reference. Earth is a rotating frame, with period of rotation equal to one day which imparts an angular velocity Ω = 0.729 x 10-4 s-1. Because the earth rotates from west to east the tendency of Coriolis force in the northern hemisphere is to rotate or steer a ‘moving body’ to its right and vice versa in the southern hemisphere. The intensity of the Coriolis force is maximum at the poles and zero at the equator. The equation for motion of planetary fluids can be summarized as (for unit mass),

where U is the total velocity, Φ is the ‘geopotential’ defined by the sum of gravitational potential and centrifugal potential of the earth rotation. For the vertical axis pointing outward along the radius of the earth thegeopotential is equal to the gravitational force of earth (i.e.). This term is approximated as to hydrostatic assumption that ∂p/∂z = -ρg.

The second term on the right hand side is the pressure gradient force. The third term is the Coriolis force. The fourth term stands for the kinematic friction and viscosities of the fluid. In the large scale planetary dynamics they represent eddy viscosity with each ‘eddy’(stands for irregular chunk of fluid) analogous to a molecule in the molecular viscosity but with extremely high mass and viscosity.

3.3 Potential vorticity

Although the momentum (and so do the angular momentum) of the planetary (or geophysical) fluids are conservative in nature, because of the rotation of the earth a virtual element of force comes into picture (Coriolis force) which often exchanges the momentum between the rotational part of the motion and the planetary rotation of the fluid inherent from the rotation of the frame of reference. In such situation a more relevant and conservative property of fluid dynamics called “Vorticity” can be defined. Due to the shear in the fluid motion the fluid elements can rotate relatively to its neighbour. This is called the relative vorticity usually measured by the shear of the flow. The equation of relative vorticity is  where U is the total velocity.

The total vorticity (which is a vector measure of rotation) is two times the local angular velocity.

The Planetary rotation (vorticity) f = 2Ωsin(θ) where θ is the latitude. Additional source of rotation comes from the gradient in the stratification of fluid across a broader horizontal distance. Because of differential heating and compositional changes of fluids in the atmosphere and ocean over a larger distance, the isobars (lines of constant pressure) and isopycnal (line of constant density) are not parallel. In this case the fluid has a tendency to make these two iso-lines parallel, which offers a tendency of rotation.

Because there are various tendencies to offer rotation or vorticity in a fluid in the geophysical fluid dynamics (GFD), a more conserved quantity can be arrived known as potential vorticity,

The potential vorticity (π) of the fluid parcel is retained over a longer period of time and space scale which offer a greater predictive value in GFD.

The above section introduced few governing rules that are handy to describe the dynamics of the atmosphere and ocean. However one should bear in mind that the solutions of GFD problems are rather intuition driven. That means when we explain a typical feature of the dynamics we only consider the most possible and necessary terms in the equation of motion, while others are neglected.

4.1     Geostrophic and quasi-geostrophic dynamics

Geostrophic dynamics is obtained just by equating two terms in the equation of motion. It is basically the balance between the pressure gradient terms and Coriolis force. In the two dimensional and horizontal cases this can be written in component form as

where u, v are the zonal and meridional velocities and p is the large scale pressure over extremely larger distances.

Geostrophic currents are time independent or degenerative in nature, meaning that if you know the pressure pattern then velocities (ocean currents or atmospheric winds) can be readily deduced from it. Since there is no time dependent term it cannot provide any information about the future of the currents. A close examination of above equations tells us that the currents are parallel to the contours of pressure as shown in the picture below.

Most of the mid-latitude atmospheric circulations are pressure pattern driven and geostrophic relation gives great amount of accuracy about the deduced winds. However in tropical regions the dynamics are not purely pressure pattern driven but convection plays an important role and therefore geostrophic relations are not very valid. In addition to that the Coriolis force which is central to the geostrophic dynamics is weaker in the tropic because of the proximity of the equator (remember that Coriolis force is proportional to sin(θ) where θ is the latitude). Hence the tropical dynamics are seldom deduced from geostrophic relations alone.

Although the geostrophic relation gives great accuracy in deducing currents or winds from pressure patterns, the geostrophic relation is degenerative in nature; meaning that it has zero predictive value because it is time-independent. Adding a time component to geostrophic relations will give a predictive value to it. This is called quasi-geostrophic dynamics (QG).

QG dynamics starts with the basic assumption that atmospheric or oceanic dynamics are not purely geostrophic but it has a slight deviation from the ‘geostrophy’. The deviations of currents from geostrophic relations are taken as u = u0 + εu1 where u0 is the pure geostrophic velocity, ε is a small coefficient representing an ‘impurity’ in the geostrophic velocities. The quantity ε is called the Rossby Number which by definition ε=U/fL, where U is the advection velocity scale, f-is the local Coriolis component and L is the advection length scale. For large L and small U, Rossby number is small (often 0.1 to 0.01). With this approximation the tendency of velocity (or expressed in terms of stream function) can be written as where, Ψ is the stream function (i.e. u = -∂Ψ/∂y, v = ∂Ψ/∂x), J is the Jacobian (i.e. J(a,b) = (∂a/∂y)(∂b/∂x) – (∂a/∂x)(∂b/∂y)), F is the Coriolis parameter, ηb is the ambient potential vorticity or the local latitude and its variability due to curvature. Readers are referred to Pedlosky(1987), and their chapter-3 for

further reading and derivation of above equation. The term  is another representation of the potential vorticity, remembering from previous section that it is a conserved quantity throughout the fluid dynamics.

4.2  Planetary wave dynamics

The solution to the above equation can be obtained by assuming a general wave solution of the form  where k is the wave number in east-west direction of the earth and ω is the frequency of the wave. Substituting this into the above equation gives a rich spectrum of waves. These are called the Planetary Waves which always dominate the atmospheric and oceanic dynamics. Away from the tropics the planetary waves are generally called as Rossby Waves. They always travel westward. Their horizontal extents are 10,000s of kilometre.

Close to the equator and tropics the planetary wave solutions become discrete in nature so that only a definite number of modes of waves exist (either symmetrical or anti-symmetrical with respect to the equator). They can be represented in a wave-number frequency diagram as depicted below.

4.4         Ekman transport

When wind blows over the earth’s surface, a stress is exerted on the surface whether the surface is solid earth or sea. In the case of ocean this stress exerts a motion in the surface layer where it is in contact with the wind stress. This is called Ekman transport. In this case we consider a balance between Coriolis force and the wind stress and its penetration with depth in the fluid medium as follows.

where, X and Y are the wind stress experiencing over the surface ocean in x and y directions, f is the local coriolis parameter and u, v are zonal and meridional ocean currents. The penetration of wind stress is typically upto upper 50 meters in the ocean. The resulting circulations are, UE = Y/ρf and VE = X/ρf, where UE and VE are the depth integrated (from surface to depth at which the wind stress effect vanishes) currents. The solutions to this equation are sinusoidal functions of depth. That means the currents has deflection from the direction of wind axis (to the right in the northern hemisphere) and it rotates with depth as a spiral. This is called Ekman spiral. The net (averaged over the depth) direction of the Ekman currents will be nearly perpendicular to the right (left) hand side of the wind direction in northern (southern) hemisphere

4.5         Ocean Gyre circulations

The ocean dynamics is established by the action of wind over the ocean. The large scale surface winds are westward in the tropics because of the trade winds and eastward in the sub-tropics to mid-latitude because of the westerlies. The Ekman transport associated with these winds pushes the water in the surface and pile them up in the centre to these wind directions. This causes a pressure gradient and geostrophic current surrounding the centre (Fig. 11). The net result is a large scale Gyre circulation in the ocean.

The solution to gyre circulation is obtained by Sverdrup theory. Consider the pressure gradient, Coriolis force and wind stress terms in the balance equation.

First take the vertical integral of this equation. Then taking the curl (i.e. of this two equations (i.e. derivative with respect to y of the first equation and subtract the derivative with respect to x of second equation) one arrives at, where, β = ∂f/∂y. The above equation means that the net north-south transport V in a Gyre is a function of curl of the wind stress.

Sverdrup equation simplifies and relates the ocean Gyre circulation to the curl of the local wind stress. However, it clearly does not depict the observed flow pattern of a Gyre circulation (Figure 11). It has broad eastern boundary stream functions indicating a broad and weak southward eastern boundary current and tightly packed western boundary stream functions indicating a narrow and strong northward western boundary current.

With sufficient modification to Sverdrup theory, and by introducing coastal boundary friction of Laplacian type (Stommel theory) or bi-harmonic type (Munk theory) one can achieve a solution to the westward intensification of boundary currents.

The explanation of gyre circulation and western boundary current intensification can be given by the principle of conservation of potential vorticity; ζ+f = constant. As water travels from higher latitude to lower latitude, the planetary vorticity (f) of it decreases (remember the planetary voriticy, i.e. f is a function of sin(θ)). But the boundary friction at the eastern boundary contributes a slight shear in the flow in the direction of local planetary vorticity and compensates the reduction in the ‘f’. Further the flow from north to south tend to have a positive vorticity (i.e. a rotation tendency in the direction of planetary vorticity and thus the stream functions of the east becomes broad and stretched.

On the other hand at the western boundary, the boundary fiction offers a shear in a positive vorticity direction (with respect to the planetary rotation). In addition ‘f’ also increases as water flows northward. Thus in order to keep the total vorticity conserved the water has to steer strongly clockwise (i.e. gain negative vorticity) right after the boundary shear. This causes intense, narrow, sheared western boundary currents (orange arrows at the western boundary).

5. Climate

5.1     Ocean-atmospheric interaction and climate dynamics

Ocean and atmosphere always exchanges mass and momentum which is the key part of the climate dynamics. Such large scale exchanges are central to the process such as El Niňo and Southern Oscillations. The intrinsic dynamics of the ocean and atmosphere as we have seen in the previous sections are key elements in the large scale dynamics of climate. A systematic narration of climate dynamics is as follows:

The large scale and differential heating from the sun warms the atmosphere by the green-house effect. This differential heating sets up the circulations in the atmosphere. Once the atmosphere is in motion, it forces the underlying ocean and induces ocean currents. The differential heating received from the sun causes the ocean to transport heat from tropics to mid-latitude and release to atmosphere there. This tends to force the atmosphere further there. The precipitation in the tropics and evaporation in the mid-latitude exchange large amount of water between ocean and atmosphere and it affects the oceanic mixing and dynamics. It takes very long time (of the order of thousands of years) for all these processes of interactions and exchange to bring equilibrium and mean state of our climate. Details of this can be found in Kump et al., (2010).

6. Summary

On the whole, the atmosphere and ocean exhibit spectacular properties of dynamics due to (a) its large size, (b) rotating frame of reference and, (c) gradient heating from the sun. The dynamics of ocean and atmosphere are integral part of the climate system and its evolution. The steady state balance in the atmosphere, ocean and climate evolutions are indeed the outcome of the intricate connections between their dynamics and exchanges of properties between them.