subjective plausibility, or the degree to which the statement is supported by the available evidence

The field draws on theories and methods including graph theory from mathematics

we need not write three laws every time we write Newton’s equations or other laws of physics. We write what looks like one law, but really, of course, it is the three laws for any particular set of axes

a⋅b=abcosθ

Vector algebra

same object, as seen from different axes. The very fact that we can say “the same object” implies a physical intuition about the reality

All quantities that have a direction, like a step in space, are called vectors. A vector is three numbers. In order to represent a step in space, say from the origin to some particular point PP whose location is (x,y,z)(x,y,z), we really need three numbers, but we are going to invent a single mathematical symbol, r\FLPr, which is unlike any other mathematical symbols we have so far used.1 It is not a single number, it represents three numbers: xx, yy, and zz. It means three numbers, but not really only those three numbers, because if we were to use a different coordinate system, the three numbers would be changed to x′x', y′y', and z′z'. However, we want to keep our mathematics simple and so we are going to use the same mark to represent the three numbers (x,y,z)(x,y,z) and the three numbers (x′,y′,z′)(x',y',z'). That is, we use the same mark to represent the first set of three numbers for one coordinate system, but the second set of three numbers if we are using the other coordinate system.

laws of physics, so far as we know today, have the two properties which we call invariance (or symmetry) under translation of axes and rotation of axes. These properties are so important that a mathematical technique has been developed to take advantage of them in writing and using physical laws

used to model pairwise relations between objects

cgs unitsof millibars (1 mb103bar)

exponentially with height

Exercise 1.1The globally averaged surface pressureis 985 hPa

atmospheric pressure is usuallyexpressed in units of hundreds of (i.e., hecto) pascals(hPa)

includingthe minus signs in front of them, are referred to asadvection terms

westerly(from the west)andeasterly(from the east)

xis distance east of the Greenwich meridianalong a latitude circle

Solution

Earth’s atmos-phere

Atmospheric scienceis a relatively new, applied disci-pline

Instability91viiContentsP732951

In 1960, the CGPM launched the International System of Units (in French the Système international d'unités or SI) which had six "base units": the metre, kilogram, second, ampere, degree Kelvin (subsequently renamed the "kelvin") and candela

The cumulative distribution function of Y {\displaystyle Y}

differentiating both sides of the above expression with respect to y {\displaystyle y}

The term "random variable" in statistics is traditionally limited to the real-valued case ( E = R {\displaystyle E=\mathbb {R} } ).

A random variable has a probability distribution

simplify

Increasing the shear depth above 4500 m reduces the precipitation again, with the values for 6000- and 9000-m depths being smaller than those for the intermediate depths.

The middepth shear layers, with tops at 3000–4500 m, produce the greatest precipitation; in this sense, these intermediate shear-layer depths are “optimal.”

The shallowest shear layer, with depth 1500 m, produces the least precipitation, about 30% less than the unsheared case, despite a greater degree of convective organization compared to the unsheared flow

two distinct regimes

the strong shear case is moister than the unsheared case by about 0.2 g kg−1 and drier in the boundary layer by about 0.5 g kg−1

improvements can make the model better suited for addressing a host of climate problems

Additional improvements in forecast skill might be possible with a state-dependent correction if the associated statistical sampling issues can be overcome (e.g., Leith 1978; Danforth et al. 2007).

There thus appears to be two windows (one early and one late) during which TBC could induce improved forecasts.

the climate drift in the North Pacific waveguide (believed to be a key controlling factor for Rossby waves entering North America) appears to develop too slowly in CNTRL-A (reaching only about one-half the long-term value at 10 days’ lead) to allow its correction in TBC-A to produce more than a modest impact (via more skillful Rossby wave predictions) on week 2 T2m forecasts (when skill is already rather low).

o improvement in the precipitation forecasts

skill improvements were rather modest at best

improved (increased) cloudiness in TBC-A (not shown) appears to contribute to the dramatic reduction in the warm bias over the NH summer continents. Here we have a clear case where the TBC impacts are indirect; the model’s parameterizations of moisture processes working with the states directly affected by TBC appear to produce more realistic output

dramatic reduction of the SST bias in TBC-C appears to be the result of a combination of direct impacts from the near-surface temperature increments (especially over the Gulf Stream, the SH high latitudes, and equatorial and coastal upwelling regions) and indirect impacts due to the reductions in surface stress biases

improvements in the equatorial surface stress

excessive subtropical westerlies in both hemispheres (though more so in the NH) and during both seasons. These likely reflect anomalous forcing/heating by the excessively strong and split ITCZ in the coupled model.

In discussing the TBC impacts on the model’s climate, it is useful to consider them as being divided into those that are direct and those that are indirect, with the latter including any quantities (such as precipitation and, for the AOGCM, atmospheric moisture) that are not explicitly forced by the TBC, as well as the transients, since the TBC is a constant forcing term.

state-independent TBC to the atmosphere can produce considerable improvements to the simulated mean climate as well as to its variability

return of skill

bias in the u wind (the waveguide) develops slowly over the course of about two months (Fig. 12, top left; green curves).

right) The υ250-mb correlations at 12-day lead for TBC-A, CNTRL-A, and the differences

how the drift in the waveguide evolves (u250 mb) and the extent to which the Rossby waves themselves are predicted more accurately (υ250 mb

hese two time scales (associated with drift development and predictability) serve to define a window of forecast leads during which TBC can be expected to have an impact on skill

skill assessment is based on a series of hindcasts initialized in late spring and running through August produced with both the CNTRL-A and TBC-A models (see section 2c). Note that in the following we use the terminology hindcasts and forecasts interchangeably, keeping in mind that these simulations are not true forecasts; in these atmosphere-only runs, observed SSTs are prescribed throughout the forecast period

correlations between the DJF mean Niño-3.4 index a

reductions are likely due to TBC-C-induced changes in the (now reduced) variability of the tropical Pacific SST linked to ENSO

substantially improves the boreal winter stationary waves (Fig. 9, right

et and stationary waves, improvements in the climatologies of those aspects of the flow should have positive impacts on ENSO-related teleconnections

ENSO-related teleconnections

does little to improve the MJO, though the CNTRL-C model already produces a fairly realistic but weaker-than-observed MJO

The improvement in the zonal mean specific humidity (Figs. 6, 7, bottom) is also substantial, highlighted by the elimination of the wet biases in CNTRL-C in the tropics and SH during both seasons (it is noteworthy that this occurs despite not correcting the moisture

we do not replay the moisture in the AOGCM

used to correct the u, υ, T, and ps tendencies

the replay approach (REPLAY-C) is able, for the most part, to reproduce the annual mean observed (Reynolds) SST. In contrast, the free-running CNTRL-C (middle-left panel) shows large positive SST biases over much of the tropics and SH

improvements are seen in the NH momentum transport, especially in the North Pacific and North Atlantic jet exit regions, where the high-frequency eddies are expected to maintain the mean jet through barotropic decay (e.g., Chang et al. 2002).

increased wet bias over India

Model changes possibly responsible for this improvement between these periods are the shift of SSTs from the use of weekly optimally interpolated (OI) SST to the high‐resolution RTG SSTs and the update of the Community Radiative Transfer Model (CRTM).

JJA averaged AIs for the years (left) 2012, (middle) 2013, and (right) 2014 at approximately 850 mb. The AIs remain quite consistent from 2012 to 2014.

Forecast surface pressure is generally too high (cool colors) over the oceans, except near coasts, and too low (warm colors) over the continents

the general applicability of the TBC approach can only be fully assessed by repeating our analysis with a number of different models (and perhaps several different reanalyses)

Additional work is needed to better quantify the statistical sampling errors as a function of season, the sizes/locations of the regions, and the length of the model runs

distinguish between proximate and ultimate causes of the model biases

the TBC impacts project less strongly on the biases, accounting for between 58% and 66% of the bias for the T2m, and between 50% and 61% of the bias for precipitation. Nevertheless, in a relative sense, considering only the fraction of the bias over North America that is actually corrected by applying TBC in northern midlatitudes, we find that the sources of the biases are again roughly 2/3 remote and 1/3 local.

eddy streamfunction response

excessive heating in that region is the main source of the AGCM’s circulation biases (and related biases) that span the NH

temperature increments at 500

reduction to the excessive precipitation bias over the Himalayas, suggesting that the associated cooling anomalies may play some role in producing the hemispheric-wide responses in those experiments

RPL_Tibet_uv) produces a more locally confined response

The remarkable similarity between the RPL_Tibet_T and RPL_Tibet_All responses indicates that the long-term bias is primarily driven by the temperature increments in that region. Somewhat similar, although with weaker amplitude, results are obtained from only replaying to the moisture (RPL_Tibet_q).

wavenumbers that are much smaller than would be expected for typical NH summer jet speeds

(Note that the calculation of the cosθ values in Fig. 11 does not include the patterns in the local region considered, since these are forced to track the analysis by design)

TBC corrects for a long-term mean error, whereas RPL effectively corrects the specific errors produced at each time step

nonlinear

TBC in TBC_TR substantially reduces the long-term JJA tropical precipitation biases (not shown), but has little impact on the NM region

2/3 from remote sources and 1/3 from the local source

local impacts tend to be more focused on the middle and southern Great Plains

remote TBC corrections provide more than twice the impact on T2M over North America as local corrections

responses are quantified in terms of a normalized spatial inner product

caused bynoise

Occam’s razor now tells us

p(r.r0jH)

The factor that updates the prior odds to the posteriorodds is called the Bayes factor

The presence of alter-native theories also influences prior odds for hypotheses

11O(H)

case of lowsignal-to-noise ratio

O(H)

p(H)

pvalue[p(r.r0jH)

Inthe limit of very low signal-to-noise ratio, the relatedseries would also show 95% low correlations and 5%high correlations (see Table 2).The probability that our observedr0withr0.rpisindicative of an actual relation is then 5/(515)550%

equal prior odds

he signal and thenoise

sampling

study synthetic time series

the errorof thetransposed conditional

fluke

depen-dent

attest) provides an answer

time series, and we find that they arecorrelated

lead

large fraction of papers in the climate literature includes erroneous uses of significance tests.

Tukey (1991), however, "It is foolish to ask 'Are the effects of A and B different?' They are always different—for some decimal place."

Trier and Parsons (1993) noted how a trough moving over the Rocky Mountains and into the Great Plains area strengthens the climatological southerly low-level jet that feeds moisture into MCSs forming over the central United States (Fig. 17-45). A similar behavior occurs in South America, where the South American low-level jet (SALLJ) flows southward along the eastern edge of the Andes from the moist Amazon region to feed MCSs in the region centered on Argentina (Nogués-Paegle and Mo 1997; Douglas et al. 1998; Saulo et al. 2000; Marengo et al. 2004; Vera et al. 2006; Salio et al. 2007; Rasmussen and Houze 2016). As shown by Bonner (1968), these low-level jets are stronger at night, which gives nocturnal preference for MCSs over the central United States (as noted by Huckleberry Finn, see introduction). Dai et al. (1999) showed how the diurnal and semidiurnal processes favor large-scale convergence over the Rockies during the day and over the plains to the east at night. These processes assure that the enhanced jet associated with an approaching trough has its maximum effect on MCSs at night in the central United States. Data from the U.S. radar network show that MCSs developing from diurnally triggered convection over the Rockies and propagating eastward maximize at night over the central United States (Carbone et al. 2002), in conjunction with the nocturnal maximum of the low-level jet in that region. Feng et al. (2016) found that an increase in MCS activity over the central United States has been accompanied by strengthening of the low-level jet and its moisture transport over the past 30–40 years.

low specific (3-6 g/kg) and absolute humidity (4-7 g/m3)

suppression will minimally requirea combination of social distancing of the entire population, home isolation of casesandhousehold quarantine of their familymembers.

we apply a previously published microsimulation model to two countries: the UK (Great Britain specifically) and the US. We conclude that the effectiveness of any one intervention in isolationis likely to be limited, requiring multiple interventions to be combined to have a substantial impact

TQV1:00Z – TQV0:00Z = 3600 (DQVDT_DYN + DQVDT_PHY + DQVDT_ANA) 0:30Z

The exergy weighting factor (20) explains the expected behavior for wq(z), which increases with height for decreasing values of rυ⎯⎯⎯(z)rυ¯⁡(z)<math altimg="eq-00078.gif"><mrow><mrow><mover accent="true"><mrow><msub><mi>r</mi><mi>υ</mi></msub></mrow><mo>¯</mo></mover></mrow><mo>⁡</mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></math>.

one of the most powerful inventions of modern science.” 1 1. J. Gleick, Chaos: Making a New Science, Viking, New York (1987). But who invented it? Who named it? And why

“phase space” has become synonymous with the idea of a large parameter set

ROJECT a single eigenvector onto the data and get an amplitude of this eigenvector at each tim

One way to represent their amplitude is to take the time series of principal components for the spatial structure (EOF) of interest, normalize this time series to unit variance, and then regress it against the original data set. This produces a map with the sign and dimensional amplitude of the field of interest that is explained by the EOF in question. The map has the shape of the EOF, but the amplitude actually corresponds to the amplitude in the real data with which this structure is associated. Thus we get structure and amplitude information in a single plot. If we have other variables, we can regress them all on the PC of one EOF and show the structure of several variables with the correct amplitude relationship, for example, SST and surface vector wind fields can both be regressed on PCs of SST

σk2=λkN

You must choose which dimension of the data matrix contains interesting structure, and which contains sampling variability. In practice, sometimes only one dimension has meaningful structure, and the other is noise. At other times both can have meaningful structure, as with wavelike phenomena, and sometimes there is no meaningful structure in either dimension

Singular Value Decomposition

The decision to standardize variables and work with the correlation matrix or, alternatively, to use the covariance matrix depends upon the circumstances. If the components of the state vector are measured in different units (e.g., weight, height, and GPA) then it is mandatory to usestandardized variables. If you are working with the same variable at different points (e.g., a geopotential map), then it may be desirable to retain a variance weighting by using unstandardized variables. The results obtained will be different

Orthogonality of the Principal Component Time Series

Equation (4.18) shows how to express the original data in terms of the eigenvectors, when the coefficient matrix Zis defined by (4.17)

orthonormal eigenvectors ei

eTXXTe

covariance matrix is diagonal in this new coordinate space

structure and sampling variables

CCA is MCA of a covariance matrix of a truncated set of PC’s

develop guidelines and best practices to educate the future Earth science workforce to be well prepared for innovative, interdisciplinary research

collaboration between Earth scientists and AI researchers

constraints in the optimization problem

a two-step approach. For a given task, first identify all subtasks that can easily and efficiently be addressed by physics-driven methods, and apply those

prediction, understanding

physics-based and data-driven methods simultaneously

e.g., empirical orthogonal function analysis and spectral analysis

Without best practices, inappropriate use of these methods might lead to “bad science,” which could create a general backlash in the Earth science community against the use of AI methods. Such a backlash would be unfortunate because AI has much to offer

atural selection is the survival of the survivors

Machine Learning for Everyone In simple words. With real-world examples. Yes, again

programming paradigm

A Tour of Machine Learning Algorithms

Why Predict? Historical ; Perspectives on Prediction in Earth Science

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less inciination to misapply evi-dence

The mindlingers with pleasure upon the factsthat fall happily into the embrace ofthe theory, and feels a natural cold-nc,s toward those that seem refractory

the method of multiple work-ing hypotheses

About the Framework

Inferring models from observations and studying their properties is really what science is about.

misregulation hypothesis proposes that individuals may instead prioritize downregulating negative emotions (e.g., anxiety) through procrastination over accomplishing

beliefs or confidence regarding one’s capability

Our mathematical nomenclature is as follows. Capitalized Latin letters denote random variables,122and lowercase versions of the same letter indicate particular values of these variables. Vectors123are boldfaced. We define ̃Yto be a climate sensitivity proxy such as the equilibrium climate124sensitivity or a climate feedback strength, for which the constraints are derived. A single emergent125constraint variable is denoted ̃X. A collection ofnemergent constraints will be labeled ̃Xi,i=1261,...,n. Versions of these random variables which have been normalized to have zero mean and127variance of 1 are similarly denoted, but without the tilde. The PDF of any random variableUis128p(u), and similarly for multivariate distributions.

community

Daubert says, the methods should be judged by the following four criteria

There should be a known rate of error

error is intrinsic to our interaction with nature

force

a list of such pretenders

data analysis and the proper application of research methods

two-thirds of all postdoctoral fellows in biology in American universities believe that they are going to make this step, but in fact, only about a quarter of them succeed

media outlets

inconsistent with human nature

choose what is and is not worth observing

we understand

cleansed of harmful preconceptions

demystify somewhat the business of science

But such processeshave not been well characterized.

Deep learning for multi-year ENSO forecasts

trivium

standard deviation σ/√n. Where σ is the standard deviation of the sample and n is the number of observations in the sample

Identifying Causal Effects from Observations

Concepts You Should Know

Reminders from Basic Probability

Scientists should also make their data and all other relevant materials available to the world once they publish their research

calculated the Jacobian using thenumdifftoolsPython package. The specific268method we used from this package is a second-order forward difference method

he proposed framework uniquely elucidates therelationship between the IT and statistical perspectives on causalit

distributions over the effect rather than values of the effect and(2) are defined with respect to random variables representing a cause rather than specific values

We conclude that there is a need to more accurately quantify entrainment rates, improve the representation of plume radius, and incorporate the effects of column instability in future versions of 1D volcanic plume models.

quasi-geostrophi

verticallypointing lidar.

Morning convection is particularly strong in the Gulfof Panama because of its concave coastline

abatic wi

ractional cloud coveragetends to be highest around sunrise and lowest duringthe afternoon. The thinning (and in some cases thebreakup) of the overcast during the daytime is due tothe absorption of solar radiation just below the cloudtops (see Fig. 4.30

evaporation of the drizzle drops in thesubcloud layer absorbs latent heat. The thermody-namic impact of the downward, gravity-driven flux ofliquid water is an upward transport of sensible heat,thereby stabilizing the layer near cloud base

In contrast, cooling from above drivesclosed cell convection

Heating from below drives open cellconvection

horizontal roll vortices

The area indicated by the hatched region representsthe total amount of heat input into the bottom of the boundarylayer from sunrise until time t1

turbulent sensible heatflux FHacross the mixed layer

evolution during summer over land

nonlocal air-parcel movement

Typical variation of wind speeds with height in thesurface layer for different static stabilities

Karmam

logarithmic wind profile is consistent with aK-theory approach